j.digital.lib.lehigh.edu/fritz/pdf/400_4.pdf · · 2012-08-012.15 subrou'l'ine...
TRANSCRIPT
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FLMXPK - A MATRIX PACKAGE
by Sampath N.S. Iyengar
Celal N. Kostem
Fritz Engineering Laboratory Department of Civil Engineering
Lehigh University Bethlehem, Pennsylvania
September 1971
Fritz Engineering Laboratory Report No.400.4
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COPYRIGHT
© 1971
SAMPATH N. S. IYENGAR
CELAL N. KOSTEM
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400.4
l . TABLE OF CONTENTS Page
] SYNOPSIS i
1. INTRODUCTION 1
] 1.1 A Brief History 1 ~
] 1.2 Divisions in the Text 2 .
2. ROUTINES 4
2.1 SUBROUTINE ADD(A,B,C,M,N) 4
2.2 SUBROUTINE DETMT(A,DA,N) 6
] 2.3 SUBROUTINE DIAG(A,DA,N) 13
2.4 SUBROUTINE EV(A,S,N) and ENTRY IEV 14
2.5 SUBROUTINE GEVP(A,B,S,T,N) 29
2.6 SUBROUTINE MINV(A,N,DET,NEXCH) 34
2.7 SUBROUTINE MOVE(A,B,M,N) 50
2.8 SUBROUTINE MULT(A,B,C,L,M,N) 51
2.9 SUBROUTINE OUTE(A,I,J,TITLE,TITEL) and ENTRY OUTF. and ENTRY OUTG ·s3
2.10 SUBROUTINE PMULT(A,B,K,L,X) 59
2.11 SUBROUTINE POSTM(A,B,K,L,X) 62
2.12 SUBROUTINE RDCBC(A,M,N) SUBROUTINE RDCOLG(A,M,N) SUBROUTINE RDRBR(A,M,N) SUBROUTINE RDROWG(A,M,N) 65
2.13 SUBROUTINE SCMUL(A,M,N,X) 67 .•
2.14 SUBROUTINE SINV(A,DA,N) 68
.. 2.15 SUBROU'l'INE SOLVE(A,B,N,L,DET) 77
2.16 SUBROUTINE SQTR(A,N) 80
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400.4
2.17 SUBROUTINE SUB(A,B,C,M,N)
2.18 SUBROUTINE THULT(A,B,C,L,M,N)
2.19 SUBROUTINE TRANS(A,B,I1,N)
2.20 SUBROUTINE XABATC(A,B,C,L,H,X) SUBROUTINE XABTA (A,B,L,N,X) SUBROUTINE XABTC (A,B,C,L,M,N) SUBROUTINE XATBAC(A,B,C,L,M,X) SUBROUTINE XATBB (A,B,L,N,X)
3.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
4.
4.1
4.2
4.3
4.4
4.5
5.
GENERAL NOTES
CO!'-ftv10N Block Labelled IYENGAR
Use of Variable (Adjustable) Dimensions
Reserved Names
Subprogram Lengths
Temporary Vectors
Features of the CDC 6400
Use of Single Subscripts in Selected Routines
USER'S GUIDE
Introduction
General Limitations
Disclaimer
Deck Setup
Description of the Subprograms
READY REFERENCE SHEET
APPENDIX 1 (Proofs of Some Basic Theorems)
APPENDIX 2 (Listing of Subprograms)
ACKNOWLEDGMENTS
REFERENCES
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100
102
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106
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107
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140
150
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J _] 400.4
1. INTRODUCTION
J 1.1 A Brief History
J The matrix package MXPAK was used in course work and
'-research by members of the Fritz Laboratory until the
]· GE 225 computer at Lehigh University was replaced· by the
·-] CDC 6400 in the Summer of 1968. MXPAK was written in
LEWIZ, a language developed at Lehigh, and was compatible
:.] for use with the GE 225 put not the CDC 6400.
To meet the immediate needs of the Laboratory,
] FCHXPK (FORTRAN Callable Matrix Package) was written by
J Mr. Edward T. Manning, Jr. and Mr. Iyengar in Fall 1968.
Later, a documentation of this package was considered
] worthwhile in view of its highly satisfactory performance.
At this stage, it became evident that many improvements .] -· and some additions were possible. The resu~t of all such
] modifications is the present version, FLMXPK (Fritz
Laboratory Matrix Package), which is described in the
"] -· succeeding pages.
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400.4
1.2 Divisions in the Text
FLMXPK contains 30 routines to perform matrix operations,
as summarized in the following table.
. Operation
Add matrices
Subprogram
ADD DETHT,MINV,SINV,SOLVE DIAG
Determinant of matrix Create diagonal matrix Eigenvalues of symmetric Invert matrix
matrices EV,IEV,GEVP
Copy matrix Multiply matrices
Print matrix Read matrix
Simultaneous equations Transpose matrix Subtract matrices
MINV,SINV MOVE MULT,PMULT,POSTH,SCHUL,
TMULT,XABATC,XABTA, XABTC,XATBAC,XATBB
OUTE,OUTF,OUTG RDCBC,RDCOLG,RDRBR,
RDROWG SOLVE SQTR ,·TRAi'lS SUB
Each of these routines has received individual treatment to
the extent considered necessary (Chapter 2). In general, the
discussion-is under the following headings:
1. Function 2. Development of the Subprogram
(including the mathematical background) 3. Special Features (if any) 4. Limitations (if any) 5. Additional Remarks (if any)
The limitations listed under each routine are in addition
to the general limitations which are discussed in Section 4.2
of the User's Guide (Chapter 4).
Chapter 3 - "General Notes" - contains remarks which
apply to the package as a whole.
The User's Guide (Chapter 4) explains briefly how each
routine may actually be used. The more important limitations
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400.4
of each routine are listed here also. The ready reference
sheet (Chapter 5) is hopefully all that a user needs in day
to-day work after he has gained some experience in the use
of this package.
Proofs of certain theorems in matrix algebra and topics
related to the text are included in Appendix 1 to. make the
subject matter as complete in content as possible.
Finally, the subprograms the~selves are listed in
Appendix 2.
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400.4
2.1
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2. ROUTINES
SUBROUTINE ADD (A,B,C,M,N)
2. 1. 1 Function
The sum of matrices A and B fs made available to
the calling program as matrix C. C = A + B.
Hatrices A,B, and C are each of size M r_ows by N
columns.
2.1.2 Development of the Subprogram
In algebraic terms, the addition of two matrices
may be expressed by CI J = AI J + BI J• From this de-, I I
fini tion, the following subprogram is e_asily written.
SUBROUTINE ADD(A,B,C,M,N) CO.tv' ... NON/IYBNGAR/I ,J, Y (12) REAL A(M,N) ,B(M,N) ,C(M,N) DO 1000 I = l,M DO 1000 J = l,N
1000 C(I,J)=A(I,J)+B(I,J) RETURN END
The use of the labeled COHHON block is discussed
under "General Notes" in Chapter 3.
In the above routine, the computations required for
locating the elements in the three double-subscripted
arrays are time-consuming. The matrices may, however,
be treated as single-subscripted arrays in the sub-
program to reduce the time required for address compu-
tations.
A further advantage of this procedure is that the
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operations can now be performed using a single DO-loop
as indicated in the listing (Appendix 2).
The size of each matrix is M*N elements but a dec
laration of the type REAL A(H*N) is invalid. This dif
ficulty is easily overcome by dimensioning each array
as a vector consisting of one element only. · When the
subprogram is CALL-ed, the starting addresses of each
array are passed to the subprogram and hence, in the
subprogram, the arrays are spaced as required in spite
of the arbitrary dimensioning.
2.1.3 Additional Remarks
The resultant matrix C can be stored in either of
the original matrices (say A). In this case, the origi
nal matrix (A) will be destroyed.
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400.4
2.2 SUBROU~INE DETMT(A,DA,N)
2.2.1 Function
The determinant of the given matrix A of size N rows
by N columns is made available to the calling program as
DA.
2.2.2 Development of the Subprogram
The basis of operations is to reduce the given mat-
rix to a lower or upper triangular matrix by elementary
transformations and form the product of the diagonal
elements of the reduced matrix to give the determinant.
In the process of such a reduction, whenever a submatrix
in the upper left of the original matrix is singular,
division by zero will be encountered. Row or column ex-
changes are necessary to overcome such a situation, if
it exists, before each step of the reduction process.
This subprogram utilizes column exchanges and re-
duces the original matrix in its own space to an upper
triangular matrix.
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Assume that the matrix has the following form at the
end of I-1 (I> 1) stages of the reduction process.
Al,l A1,2 . . . Al,N
0 A2 2 . . . A2,N 0 0 ' 0 0
0 0 AI,I AI,I+l AI,N
0 0 AI+l, I AI+l,I+l• AI+l,N
.
0 0
Except possibly for the elements in row 1, all the
elements are modified by the reduction process and, there-
fore, do not correspond, in general, to those of the ori-
ginal matrix. The part product of the diagonal elements
AK,K where K ranges from 1 through I-1 is assumed to have
been computed and stored in DA.
Further processing starts with row I. Since AI,I may
be zero, the immediate step is to search the Ith rmv for a
non-zero element in columns I through N. If AI,I itself
is non-zero, the process of reduction may be continued.
However, it is advantageous, in the interest of accuracy,
to choose AI,I such that it is as large as possible in
absolute value, since division by AI,I is later involved.
Hence, the Ith row is searched for the largest absolute
valued element in columns I through N.
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If all the elements searched are zero, the conclusion
is that the matrix is singular, and hence DA has the value
zero. Control is returned to the calling program in such
a. case.
Otherwise, the largest absolute valued element in row
I is located in, say, column N where N > M,;::: I. If H equals
I, the largest absolute valued element is AI,I itself and
no exchange of columns is involved. If M does not equal I,
columns M and I are exchanged, and the corresponding change
in the sign of the determinant is accounted for. It is
useful to note that such an exchange may be limited to
elements in rows I through N only, as elements in rows 1
through I - 1 do not affect the evaluation of the determinant.
For further discussion, the form of the matrix noted
above remains valid, although the values of some of the
elements may have changed because of column exchanges.
The part product of the diagonal elements is then
modified to include AI,I· In FORTP~N language,
DA = DA*A(I,I).
The next step is the process of reduction. To reduce
the elements in column I of rows I + 1 through N to zero, -A
it is necessary to add !1 ·I times the Ith row to row H, AI,I
where (I + 1) $.. H.:S N. After such additions, the modified
matrix has the following form.
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""' 0·
Al,l A1,2 • Al,N 0
"'"' 0 A2,2 • A2,N
0
0
0
• 0 AI,I (Elements in this row need not be modified)
0 0 0 0 0 0 AI+l,I+l-Ar+l,I
AI,I+l AI+l,N-AI+l,I
Ar,N Ar,I Ar,I
0
0
I • 0 \.0
Ar-1 I AM, I I
0 0 0 0 0 0 AH, I+l ' AI,I+l AM,N AI N Ar,I AI,I . I
0
0
0
0 0 0 0 0 0 ~,I+l AN,I
AI,I+l AN,N AN,I
AI,N A AI,I I,I
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Since the purpose here is to evaluate only the
determinant, better speed can be achieved by avoiding
generation in the machine of fresh values for some of
the elements. Some of these values are known (zeroes
below AI,I in column I) and others do not affect the
final product (elements to the right of AI, I- in row I).
The general formula, then, in the region of interest,
is -simply AH,I
AM,J = AH,J -AI,I
AI,J ,
where both !1 and J range in values from I + 1 through N,
and the prime* denotes the fresh values computed for
the corresponding elements.
AH I The elements are evaluated row by row and ---'- is
AI,I recognized as a constant \vhen elements in row M are
processed.
When the search and reduction process explained
above is repeated for a total of N-1 cycles, the matrix
is (upper) triangulated and all that remains is to mo-
dify the part product of the diagonal elements obtained
thus far by multiplying it by AN,N to give the final
value of the determinant DA.
*In the subprogram, it suffices to redefine AH J as the quantity on the right-hand side of the equati~n.
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,.,
A special situation arises if N is 1. The normal
reduction process in this case would imply operations
in stages 1 through 0, which is meaningless. However,
. the determinant is simply A1 , 1 of the original matrix
(1 by 1). Hence, very soon after the subroutine is
entered, N is checked for equality with 1 and if the
test is true, the determinant is set to A1,1 and con-
·trol is returned to the calling program.
2.2.3 Features
The operations involved are carried out within the
space of the given matrix itself. Thos~ operations
which do not affect the final result are avoided
altogether to ensure better speed of performance.
2.2.4 ·Limitations
The original matrix A is destroyed in the process
of evaluating the determinant~
It is conceivable that the determinant of a matrix
which is singular may not be identically zero when this
subprogram is used, because of machine errors in using
floating-point arithmetic. A check of the diagonal
element AI,I for its absolute value at each stage of the
reduction process (after column exchanges, if any) with
a number like 1.0 x 10-6 could have been employed to
recognize the singularity of the matrix, but this has
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an obvious disadvantage. The determinant is a function
of the values of the elements in the original matrix as
well as their positions. Hence, a purely arbitrary num
ber like 1.0 x 10-6 is inadequate for such a test. In
most engineering applications, the user has a feel for
the value of the determinant based on the trend of a
set of calculations and is normally able to recognize a
singular matrix when the determinant is "small".
2.2.5 Additional Remarks
If the value of DA is to be printed, it is better to
output this quantity in E-format (or G-format) than in
F-format since, generally, its magnitude is unknown.
This is the only subprogram (in this package) that
can also be written as a FUNC~ION subprogram [FUNCTION
DA (A,N)]. Were it so written, it would have to_be
"referenced" instead of being "CALL-ed" by a calling
program. In its present form, no exception is necessary
to the general rule of CALL-ing any of the subprograms.
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] 2. 3 SUBP.OU'r IL-!E D lAG (l\, DA, lJ)
2.3.1 Function
J. Ilatrix A of dimension N rm·1s by ;:~ columns is gene-
J rated as follows. All the diagonal elements have the
same value DA, and the off-diagonal elements are all
] zero (Diagonal matrix).
~1 .,
2._3.2 Development of the Subprogram
Treating the matrix as a vector of N*N elements in
"j the subprogram, a null array is created in the first
l DO-loop (Appendix 2). Next, the diagonal elements are
each assigned the value DA in the second DO-loop.
] 2.3.3 Additional Remarks
"] DA should be defined as a FOETl\.2\N real nu:nber in
the calling program.
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2.4 SUBROUTINE EV(A,S,~) and E~TUY IEV
2. 4. 1 FuncJcion
Eigenvalues and eigenvectors of the symmetric matrix
A are computed.
2. 4. 2 Development of the S1..1.bprogram
A. Theory and a Practical Approach
The procedure presented here is based upon the
mathematical discussion of the problem outlined by
(1) Greenstadt .
If X is a scalar and X is a (non-zero) column vee-
tor such that AX = A. X, vector X is known as the eigen-
vector corresponding to the eigenvalue X • c. ; . -,)lnce ma-cr1.x
A is of size N (N rows by N columns), it ha~ N eigenvalues
and N eig~nvectors. Since matrix A :i.s also ~~ymmetric,
its eigenvalues are all real but not necessarily distinct.
However, N distinct eigenvectors can b~ found. In fact,
it can be shown that these eigenvectors are also orthogo-
nal; that is, mutually perpendicular to each other.
] If the eigenvalues represent the diagonal elements
]
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of a diagonal matrix D, and the corresponding eigenvec-
tors (normalized) represent the columns of a matrix S,
the equation AS = SD tepresents a generalization of
AX = XX. (Recall that postmultiplication of ~atrix S -(
by a diagonal matrix D is equivalent to multiplication
of each column of matrix S by the corresponding diagonal
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400.4
. .element of matrix D). Since the X vectors are crtho-
normal 1 s iS an OrthOn0l"n1dl m.J.triX an<..t hence sri.' ::: s-·1,
The equation AS = SD can, therefore, also be written as
In mathematical terms, if S is an orthonormal matrix
such that the transform (sTAs) of matrix A is -a diagonal
matrix, the diagonal elements of the dia."gonal mu.trix are
the eigenvalues of matrix A and the columns of natrix S
are the corresponding eigenvectors. The requirement is,
therefore, to find tha matrix S. In the case of a 2 x 2
matri~, it can be found almost directly, as shown below.
A = t:·: L ,
Hatrix A is syfiu:tetric.
Assume S = [ co~ e sin e ]
--s ln e cos 6
Matrix S is orthonormal, and the value of e is to be
found. · The transform of h is given by s'l'As 'dhich is
symmetric.
= [A1 , 1 cos 2e + A 2 ,~sin2e..:. 2A1 , 2 cosesinG
(Al,l-A2 , 2 )cos8sin8 + A1 , 2 (cos2e-sin2e)
l ( ) . ( 2 . 2,\1 A1 , 1 -A2 , 2 cos9sln8 + A1 , 2 cos 8-sln 8,
1
A1 , 1sin 2e + A2 , 2cos 2e + 2A1 , 2sin9cos6 J
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Hence, A will be diagonalizcd if 8 satisfies the rcl2tion
(A1 , 1-A2 , 2 )sin 28 = -2A1 , 2cos 28
-2Al 2 I '
or tan 28 = ( 1)
A simple application in Civil Engineering is the eva-
luation of principal stresses in an element which is
ject to norr7lal and shearing stresses. I (5 = 6 l yy
_, ______ --- C5 = r3 ,--------, ~ 1/
1' I xy ' 1
~1-- : i --;-- C5 = 4
li I v XX L_ ____ __J
------;r
I t
Figure l(a)
sub-
The stress tensor for the loading shown in Figure l(a) is
From Equation (1), tan 28 -4-6
Hence, the principal stresses are given by the diagonal
elements of the follm1ing matrix:
[
3. 0
0.0
0. OJ 7.0
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The or-ientations of ti1e pr·incip2..l pl2..~1e s are c:cpi cted
in Figures l(b), l(c), and 1(d).
(0 '0)
Minor-
-~ (J
I YY = 6
I ~--- (J = ./3
\ II_:,_ y'/" \ /
,._- \ .:-;·~\ .
0
30
Fig. 1 (b)
(J
XX = 4
Fig. 1(d)
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Fig.· 1 (c)
Prin.Pl.
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When matrix A is larger than 2 x 2 it is not possible
to findS directly, since there is more than one of£-
diagonal element to be annihilated. ['i"lhenever an off-
diagonal element is referred to below, it is assumed to
be in the upper half of the matrix (subscripts P,Q where
p < Q) ~ By symmetry, there is a corresponding- (Q,P)
element identical in value in the lower half. Both
elements are affected in the same way t:1roughout.]
However, it is possible to annihilate a selected off-
diagonal element Ap,Q if matrix A is transforoed by R,
where R is the Identity matrix modified with regard to
the following elements only.
RP,P = cos e
RQ,P =-sin 9
Rp,Q=' sin 8
RQ,Q= cos e
t•latrix R is orthonormal and tan 29
(See Appendix 1 for the values of sinS and cose from this
equation). In the transform RTAR represented by matrix
B, BP,Q (and BQ,P) will be zero. We now need to continue
this process to annihilate some other off-diagonal ele;nent
of matrix B by a transform using another orthonormal
matrix, say T. ,.,
( RT ) '-A ( ET ) .
Since R and 'l' are both orthonormal, R'r is orthonormal
(Appendix l) . In other words, Cis the transform. of A by
RT, although it was shown to be derived in two steps
using elementary orthonormal transformation matrices R
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400.4 and T in turn.
With the transform of B to C, usually it happens
that BP,Q' whi~h was zero, will not retain that value.
The annihilation of AP,Q is therefore undone.
Naturally, therefore, the question arises whether a
process such as this is convergent. The answer in the
affirmative was first provided by Jacobi. (The ?roof
is not presented here.) To achieve rapid convergence,
he also proposed selection of that off-diagonal element
which has the largest absolute value, for annihilation
at the subsequent step of the process.
The selection of the off-diagonal element may be
modified slightly when using the high-speed electr.onic
computer(l). The procedure is as follows.
The square root of the sum of the squares of the off-
diagonal elements in the upper half of matrix A is calcu-
lated first. (Note that squaring each element emphasizes
the importance of the elements having larger absolute
values. The principle is analogous to that used in Least
Squares Method for Curve Fitting.) This is called the
initial threshold (TURI in the subprogram) . If diagonali-
zation of A can be continued indefinitely to the point
where all off-diagonal elements are zero, the final thres-
hold will be zero. For practical reasons, it is necessary
to terminate the program when the absolute value of each
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off-diagonal element is sr.taller thc:..n a specified valur.:;
(THRF in the subprogram) . Since the magnitudes of the
original off-diagonal elements will differ with each
problem, it is further necessary to relate THRF to these
elements in some way. This subprogram assumes that 'riiRF
is one-millionth of THRI. (Greater accuracy ~an be
achieved as discussed later under ENTRY IEV.)
Having set the value of THRF, the value of THRI 1s
reduced by dividing it by a number Hat least equal to N,
the size of the matrix. In this subprogram, N itself is
the nurilier chosen. 'l'hen, there is at least one off-
diagonal element which is larger in absolute value than
this new value of THRI{l).
Any off-diagonal eleffient which exceeds THRI in
absolute value is a "candidate" for annihilation. The
choice is made syste;:natically. If the first of£-diasonal
element (A1 ,2) is a candidate, it is annihilated.
Al,3 is considered.)
(Else,
Assuming A1 , 2 was annihilated in the previous step,
the next off-diagonal element considered is Bl,3· (Note
that A is transformed by now to B.) The next in order is
c2 , 3 , etc. (column by column).
At the conclusion of the first sweep, there is no
guarantee that every off-diagonal element \·!ill be less
than THRI in absolute value, since annihilation of an
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l element in a particular step may have been undone by sue-
l ceeding transformations. This is no serious problem since
one or more repeat sweeps will assure that THRI is larger
l- than every off-diagonal element in absolute value, by
virtue of convergence of the process. The necessity or
)· otherwise of a repeat sweep is indicated by the value of
J the logical variable IND in the subprogram.
THRI is now further reduced by dividing it by N and
l the process repeated all over again. When THRI, so
modified successively, becomes less than THRF, the diagona-
l lization is terminated.
1 B. Programming De·tails
7he procedure calls for many matrix multiplications.
-~ -· Although several names have been used in the earlier dis-
J cussion for the sake of clarity, the subprogram actually
uses only matrices A and S. The original matrix A is con-
J tinuously modified so that on return to the calling program,
J it is diagonalized and has, for its diagonal elements, the
eigenvalues of the original matrix A.· Matrix S is ini-
j tially defined to be the Identity matrix and is also
modified at each step so that finally it will store, in
J . its columns, the eigenvectors of the original matrix A.
_ .. , -·"
Assuming calculatio~Bat stage J have been completed,
the modifications required for matrices A and S at stage
J J + 1 will now be noted. Let R be the elementary
J -21-
.-1
l 400.4
l orthonormal transfor::c1ation matrix during this stage. For
'l convenience, again, let B = RTAR and T = SR.
Since matrices A and B are synm1etric and R contains a
r nuwber of zero elements, it is reasonable to suspect that
regular matrix multiplication is not warranted. A long-
.1 hand matrix multiplication shows that the following equa-
., tions are true. These may, however, also be derived .
~ (AP,Q is the element to be annihilated.)
l BP,K = A cose - AQ -,. sine where K :j p or Q (2) P,K ,h
l BQ,K = Ap T' sine + AQ,K cose where K _J, p or Q ( 3) ,t>- I
BI1
P = AI P cose - AIIQ sine where I t- p or Q ( 4) I
1 BI,Q = AI P sine + AIIQ cose where I =J p or Q ( 5) I
']
l
B = AI,K vlhere I ancl. K are both different ( 6) I 1 K from p and Q
2 sin2e sinecose ( 7) Bp p = Ap p cos e + AQ,Q - 2Ap
1Q
I I
BQIQ = Ap p sin2e + AQ,Q cos2e + 2AP1Q sinecose ( 8)
'
.1 BPIQ = B = (Aplp - AQ,Q) cose sine + A0 Q(cos2e-Q,P .. '
sin2e) ( 9 )
.1 TI1K = SI K where K t- p or Q ( 10)
'
.1
TI,P = si,P cos8 SI Q sine ( ll) ' - .
TI,Q = si P sine + si,Q cose (12) I
l This list of equations enables us to recognize a few
l features which, when incorporated in the program, speed up
operations considerably.
l :l
-22-
l
400.4 The matrix B is the s.:rrne as matrix A, except fer ro'.-:s
P and Q and colunms P and Q. Further, by virtue of s :;·m;~>e try,
.] rows P and Q are the same as colunms P and Q, respectively.
]. The following scheme to generate m~trix B (the new A) can
therefore be adopted.
. . -]- l. Calculate a new value for each element- of column P .
f-lodify the corresponding element in row P to have
the same value.
Do the same for column Q and row Q in the same DO-
loop. Equations 4 and 5 are to be used.
-·]. -·
2. As a consequence of step l, we have some fictitious
values for (the pivotal) elements Bp1
p 1 Bp,Q, BQ,P
and BQ,Q as these elements also have been conputed
using either Equation 4 or Equation 5 .
. this ">vasteful" set of calculations (the values \
] will be discarded) has saved any checking of S'lJb-
scripts that would otherwise be involved in Step l.
J The elen-:ents of the pivotal set are nm-1 evaluated
according to the Equation~ 7, 8, and 9. The
"mistake" is thus corrected.
] Turning next to matrix T, it is seen that, again, only
columns P and Q of matrix T are different from those of
matrix S. The calculations needed can be easily cornbined
1 with those in Step l above.
1 -23-
l
l
l 400.4
1 C. ENTRY IEV
It was remarked under Part A in the above discussion
1 that the accuracy of calculations need not be confined to
the degree where THRF equals one-millionth of the initial
l threshold value. The following explains this feature.
1 The diagonalization of matrix A was shown to be an
iterative process. Since a limit to this process was re-
quired to be set for practical reasons, THRF was set ·to the
value stated. The fraction, one-millionth, was arbitrarily
chosen assu:raing that the requirements of the user are there-
l by met. (Reference 1 uses one-billionth in a numerical
example.) If, however, a user is interested in greater
l accuracy, all that is necessary is to return to the sub-
l program from the calling program and thus continue the
process which is iterative. In other words, the subprogram
:I should be CALL-ed again. However, one snag of this pro-
cedure must be avoided .
. I If the subprogram is CALL-'ed again in the fo1.·m CALL
'1
EV(A,S,N), the S-matrix will get reinitialized to the
Identity matrix. So, although the eigenvalues will be
. l improved, the eigenvectors wi 11 all have wrong. valu.es .
The ENTRY statement is useful to avoid this difficulty. 1
.j The statement ENTRY IEV in the subprogram allows an
1 alternate point of entry (the next executable statement
I after the ENTRY statement). Hence, if the CALL statement
l l
-24-
J
-··j j
:J ,
j
l
.1
l j
400.4
is CALL IEV(A,S,N), the initializa~ion of the matrix S
will be bypassed. Thus, matrix S also will be improved.
In the subprogram, the positioning of the ENTRY state-
ment is such that calculations for THRI are also bypassed.
This was deliberately done for the following reasons.
With the first use of the subprogram, the ~alculations
for the initial threshold will be made and the diagonaliza-
tion procedure assures us that every off-diagonal element
is less than one-millionth of this quantity in absolute
value. The variables THRI and THRF in the subprogram,
defined prior to the ENTRY statement, represent these quan-
tities. The variable THRI changes in value during the
execution of the subprogram a~d when the diagonalization
is terminated, it is less than THRF. If now, THRF is re-
duced to one-millionth of its mm value, we will have set
the final threshold for a subsequent use .of this subprogram
and be ready for the improvement procedure, without having
to calculate a new initial threshold. (See the redefini-
tion of THRF just prior ·to the RE'l'URN Statement.) The accu-
racy prescribed for the second ruri would, therefore, be
lo- 12 times the initial threshold value calculated for the
original matrix A. This process is also iterative and
-6n hence in general terms, the accuracy is 10 times the
initial threshold value, where n is the number of times
this subprogram is CALL-ed.
-25-
1 ]
l
1 ]
1 1
1
l
]
1
]
400.4
The advantages are these: 1. The user has a measure
of the accuracy based on the nurrtber of CALL s ta ternents.
2. The calculat~ons for initial threshold are limited to
the first set. 3. These calculations can be combined
with the initialization of the S matrix (to the Identity
matrix) in the same DO-loop nest.
D. CALL-ing Procedure
The following rules apply to the CALL st~tements for
the reasons stated.
1. CALL EV(A,S,N) must be used the first time around.
This permits matrix S to be initialized and the initial
threshold to be evaluated.
If the accuracy prescribed is considered good
enough, the problem is solved on return to the calling
program.
2. CALL IEV(A,S,N) must not be used the first time
around, since matrix S and the initial threshold will not
be properly initialized.
3. CALL IEV(A,S,N) may be used as often as necessary
subsequent to CALL EV(A,S,N).
Trail runs have shown that the eigenvalues '.vill not
differ appreciably when improvement is attempted. This
is because the eigenvalues found in the initial run are
already very close to the exact values.
-26-
]
l j
]
]
1 ]
l .]
1 _]
]
J
J
J
l .]
.l
1
400.4
The eigenvectors shm: some improverr:ent \·lith eac:1 sub-
sequent CALL IEV(A,S,N). In engineering problems, one
improvement cycle may be considered the maxir.mm necessary.
(A good check on the final resul~s is to form the
matrix product SAST, where s is the matrix of eigenvectors
and A is the matrix of eigenvalues, and compare this Hith
the original matrix A. Since the original matrix A is
destroyed when this subprogram is used, it needs to be
saved or printed earlier for comparison if this check is
deemed necessary.)
A repeated number of improvement cycles indicates
fastidiousness with respect to accuracy uncalled for in
engineering applications. Further, there is a good chance
of creating an underflow in the machine (extremely small
quantities in absolute value other than zero itself can
not be handled by the machine).
For obvious reasons, matrices A and S should not be
affected in any Hay in the calling prog-ram betv1een t>.-10
successiva CALL-s·to this subprogram.
4. CALL EV(A,S,N) must not be used except for the first
time. The reason, as already stated, is that such a CALL
would destroy the S-matrix previously found by reinitial
lZlng it to the Identity matrix.
-27-
400.4 E. Special Case of l x l Natrix
If N equals l, there are no off-diagonal elements.
For this case, the eigenvalue is the element itself and
the eigenvector is just 1.0. In the subprogram, therefore,
N is checked for equality with 1. If N = 1, the relations
mentioned are established directly.
2.4.3 Features
Memory requirements are minimal, and speed is ensured
by avoiding full-scale matrix multiplications.
An improvement of the eigenvalues and eigenvectors is
easily possible, at the user's option, by one or more
additional CALL statements.
2.4.4 Limitations
The matrix whose eigenvalues and eigenvectors are to
be found must be syrnmetric. It will be destroyed in the
subprogram.
l J
-28-
'l -~
J .
l
.. J
..)
,.,] ·-
.]
~]
400.4
2.5 SUBROUTINE GEVP(~,B,S,T,N)
2.5.1 Function
Eigenvalues and eigenvectors of the syn:ur.etric
matrix A where AX = A.BX are computed (General Eigenvalue
Problem) . In this equation, X is the eigenvector cor-
responding to the eigenvalue A. \·lhich is a scalar quanti-
ty. B is a synune·tric matrix which is also positive
definite.
2. 5. 2 Development of the Subprogram
Iri tetms of matrices, the above equation may be
rewritten in the form
A·~= B~L
where ~ represents the matrix of eigenvectors; that is,
q? = [xl x2 X l • • • :;rj and
xl' X2' ••• 'XN are the N colunm vectors corresponding to
A. 1 1 'A 21 ••• , A.N Hhich are the N eigenvalues. L is the
matrix of eigenvalues and is a diagonal matrix.
At first sight, it would appear that the solution
may be obtained directly by writing
or C Q = qj L
and finding the eigenvalues and eigenvectors of the
matrix C, by the use of another subprogram. This would
be in order provided the subprogram· used can handle non-
synunetric matrices. This package does not contain such
-29-
I
I
''l
., '1
'1 I
400.4 a subprogram.
Since B is symmetric, B-l is also symrr:etric U\ppen
-1 dix l). The product C = B A is, however, not neces-
. sarily syr.unetric, although both B-l and A are syrmnetric
(Appendix l) .
The problem needs therefore a modification of the
matrices such that a symmetric matrix will result and
SUBROUTINE EV(A,S,N) may be used.
The matrix B can be expressed in the form B = SDST
where D and S are the matrices of eigenvalues and eigen-
vectors respectively of matrix B (Section 2.4.2 D).
Hence, A ~ = B ~ L = SDST ip L or
Since matrix B is positive-definite, its matrix of
eigenvalues (diagonal matrix) will have, for its diago-
nal elements, only positive terms and hence it is pos-
sible to express matrix D as the product of two (diago-
nal) matrices G and G, where
GI,I =ID1 , 1 and GI,J = 0.0 for I 1 J,
without the necessity of dealing ,,vi th imaginary nurrcbers.
Hence I sTA ip = DST <.P L = GGST .p L
or
which can be written as
= GST <.P L
Using the notations, ~ = GST~ -1 , F = SG and hence
-30-
--1 .: ......
400.4
M1 ~-\
~--1 ~--
~-l .' --\.:
~-l n~. :~ w •
r·1 , __
~-] ~~
, .. ] ~-·
J ... 1 ·-] -·
. J ]
_]
] .
·-] -·
J .]
']
rr F AF'l:' = '±' L
Since (FTAF)T = F1,AF by virtue of syrmnetry of A,
FTAF is symmetric. Letting FT AF = C, C '±' = '±' L, Hhere
C is synunetric.
This equation is hence in a form suitabl~ for the
use of SUBROUTINE EV(A,S,H).
The matrix of eigenvalues, L, obtained in such a
manner needs no modification as it is the same L in the
original equation A Q = B .P L.
From \J/ = GST~ the matrix of eigenvec-tors Q is J.
given by iQ = SG-l '±'
The procedure, in brief, is to modify matrix A and
obtain matrix C first. The eigenvalues of matrix C are
the required eigenvalues. The eigenvectors of matrix C
(matrix '±' ) are to be modified by premultiplic2tion by
. -1 the matr1.x SG .
-1 The importance of generating SG and saving it for
all modifications at further stages should be noted.
'i'he product SG-l is simple to obtain. -1 Denoting G as
H, and SG-l = SH as T, the elements of T are
N TI,J = E si,KHK,J
K=l -1
Since G or H is a diagonal matrix and hence
HK,J = 0 if K f J,
-31-
J
-..,1 ~ ... J
, .. , ,.-.]
-.~.,
..... ]
... J ... J
J
]
]
1
1
1 l
1
400.4
The columns of matrix T are thus the columns of
matrix S, each multiplied by a different scalar quantity.
-1 The product SG (=T) can be stored in matrix S itself.
The required modifications may be outlined in the
following manner, from the programining viewpoint .
Step 1. Obtain the eigei1values and eigenvectors of
matrix B by using SUBROUTINE EV(B,S,N). Matrix B now
stores the eigenvalues of the original matrix B and S
is its matrix of eigenvectors. (These correspond to
matrices D and Sin the derivation.)
Step 2. Obtain the product SG-l and store it in matrix
S. Note that each diagonal element of G-l (off-diagonal
elements are zero) is merely the reciprocal of the
square root of the corresponding diagonal element of
matrix B.
Step 3. Obtain the product AS and store it in matrix
Busing SUBROUTINE MULT(A,S,B,N,N,N).
Step 4. Obtain.the product STB using the property of sym
metry of the resulting matrix and store it in matrix A.
The product corresponds to matrix C in the derivation.
Step 5. Obtain the eigenvalues and eigenvectors of
matrix A using SUBROUTINE EV(A,B,N). Matrix A now cor-
responds to matrix L, the matrix of eigenvalues in the
derivation (no further modification required for matrix
-32-
r1 LJ
'111 !:,
'·~
J ]
"''] --
]
]
]
]
_]
400.4 L). r·latrix B corresponds to matrix ·y ln the derivatio!1.
Step 6. Obtain the product SB by using SUBF<.OUTDJE POSTt-1
(S,B,N,N,T) where T is a vector required in that sub-
. program for computations. £·latrix B now stores the re-
quired eigenvectors (corresponding to matrix ~ in the
derivation).
2.5.3 Features
This subprogram uses other subprograms developed in
this pc>.ckag·e, and coding is mainly a set of CALL s·tate-
ments. Additional storage spaces in the form of an
extra matrix and a vector, which are required to carry
out the computations, are believed to be the minimrn1
necessary.
2.5.4 · Limitations
Original matrices A and B must be symmetric. Matrix
B must also be positive definite. The original matrices
are destroyed when this subprogram is used. Other sub-
programs of this package as detailed in the User's Guide
(Chapter 4) must be available and loaded v-1hen this sub-
program is used.
-33-
J l .
~ ii;J
, ;.,:,3
1 . , -~~
]
l ]
]
]
l
400.4
2.6 SUBROUTINE MINV (A,N,DET,NEXCH)
2.6.1 Function
The matrix A of size N rmvs by N columns is inverted
and the inverse returned to the calling program in A it-
self. Also, the determinant of the original matrix is
computed and returned in DET.
2.6.2 Development of the Subprogram
A. The Core of the Subprogram
By a process of elementary transformations (on
rows) of A, it is possible to reduce matrix A to the
Identity Matrix, referred to as matrix B below.
Hence,
'I'ln.1s, the same elementary transformations (on rm1s) of
B will generate A- 1 .
In the discussion, the elementary transformations
will initially be limited to (i) division of a row by a
scalar and (ii) addition of a multiple of a row to
another row. Later, they will also include row exchanges.
In general, there is no need to follow any definite
pattern in the reduction of matrix A to matrix B. The
rules below, however, provide a systematic method of
ach~ving the reduction, with a view to easy programming
on a computer.
-34-
~
l l ~·, ~~· ~ -
I1 ~1 -~1
~l .. , ··-4
-~,
"~
400.4
Step 0. Create the Identity Matrix
Step 1. Let K = 1
Step 2. The pivot element is the diagonal element in
row K of matrix A. Divide the Kth row of both
matrices A and B by the pivot element. The
element A(K,K) will then be reduced to 1.0.
Step 3. Add suitable multiples of the (modified) Kth
row of matrices A and B to all other rows of the
respective Qatrices to generate 0.0 in the Kth
colurrm of matrix A. If the rm·l number is I,
-A(I,K) is the suitable multiple.
Step 4. Increment K by 1.
Step 5. If K exceeds N, stop. Otherwise, return to
Step 2.
'l'he numerical example below illustrates this proce-
dure and provides as well some useful material for later
discussion. 'I'he matrix on the left is A and that on the
right is B.
Step 1. K = 1
4.0 1.0 0.0
4.0 3.0 1.0
1.0 1.0 1.0
1.0 . 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
(The significance of the pair of dashed lines
will be explained later.)
-35-
l
l
1
1
}
]
l J
1
1 1 1 1 l 1 l
400.4
Step 2 .
l..
(
Step 3.
Step 4.
Step 5.
Step 2 .
Step 3 .
Step 4.
Step 5.
Pivot element has the value 4. 0 . Divide
rmv l by 4 . 0 . l....
..... o. ?5·> 1.0"> 0.25 0.0 0.0 0.0 ..-...-__ .....
4.0 3.0 1.0 I 0.0 1.0 0.0 I I
1.0 1.0 1.0 i I 0.0 0.0 1.0
Add -4.0 times row l to row 2' -1.0 tir.1es row
l to row 3.
ILO i 0 25 I • 0.0 0 25 I • I 0.0 0.0
I 1.0
I 1.0 0.0 0.0 I 2. 0 -1.0 I
I I 1. o I I 0. 0 l 0. 7 5 1.0 1-0.25 I 0.0
I I
K = l + l = 2
2 *' 3' Return to Step 2.
Pivot element has the value 2 . 0 . Divide ro'd
2 by 2. 0.
11.0 ! 0 25 0.0, r 0.25 I
0.0 " . t__O.O ......
1-o.s ......
.... ' I o. o 1.0> 0. 51 0.5> 0.0 / " /
_, I r I r 0.0 10.75 1. 01 t-0.25 I 0. 0 1. 0 I
I
Add -0.25 times rov1 2 to rmv l,-0.75 tur,es
row 2 to row 3.
11.0 0.0 -0.125 0.375 I -0 .125 I 0.0
I ·o. o 10.0 1.0 0.5 -0.5 0.5
lo.o 0.6251 I
0.0 0.125 -0.375 I 1.0 I
K = 2 + l = 3.
3 } 3. Return to Step 2.
-36-
I 1
l
.l
.. 1
l
1 ]
J
]
] .
]
]
]
1
400.4
Step 2. Pivot element has the value 0.625. Divide
row 3 by 0.625. ·
1.0 0.0 I 1-0.125 0.375 I -0.125 I 0.0 I I
0.0 1.0 I
-0.5 0.5 ~ 0. 0 l.. 0. 5 ..... ...
' ... 0.0 0.0 1.0> 0.2 -0.6 1. 6>
/ / /
_, .-· /
/ ,.... Step 3. Add -(..:..0.125) tirr,es row 3 to row 1, _-0.5 times
row 3 to row 2.
1.0 0.0 0.0, 1 o. 4 -0.2 0.2
i 0.0 1.0 0.01 :-0. 6 0. 8 -0.8
I I 0.0 0.0 1.0 I ! 0.2 -0.6 1.6 I
Step 4 . K = 3 + 1 = 4
Step 5. 4 > 3. Stop.
Evidently, the process has to be repeated N times.
The procedure is surnrnarized in the program section
belov7.
CALL DIAG(B,l.O,N)
DO 1000 K = l,N
TEMP = 1.0/A(K,K)
DO 2000 J = l,N
A(K,J) = A(K,J)*TEMP
2000 B(K,J) = B(K,J)*TEMP
DO 3000 I = l,N
IF(I ·EQ· K) GO TO 3000
1'El'1P = -A (I I K)
DO 4000 J = l,H
A(I,J) = A(I,J) + TEMP*A(K,J)
-37-
1
1
1 ]
l l 'J
l ]
1
l ]
1 1
1 ]
1 ']
'1
400.4 4 0 0 0 D (I I J) = 13 (I I J) + TEr1P * B ( K I J)
3000 CONTINUE
1000 CONTINUE
The program requires that i) separate memory space
for the matrix B be provided and ii) matrix B be defined
initially as the Identity Matrix. The ensuin·g discussion
aims at proving that these requirements need not be met.
Note that, in any problem, the elements that do not
lie between the dashed lines will renain the same. These
invariant elements need not be stored, if the program can
account for them suitably when required. Consequently,
only matrix A is assumed to be available. The rules
earlier stated need to be modified to suit the new
situation.
Step 0. Let K = 1
Step 1. Assume that a column vector corresponding to
column K of the Identity matrix is attached to
Step 2.
the right of the matrix.
The pivot elernent is the first element in rmv K.
Divide the I\th rmv by the pivot element and
store each element in the column to the left of
its original position. This column "shift" has
the effect of "pushing" out the element referred
to as the pivot elem.ent (to the left) and "bor
rowing" an element from the Identity Matrix
-38-
l '1 -~
: .. !
.. )
..
.]
J ]
_]
]
l ]
l
400.4
(from the right). It is helpful to visualize
that the kinks in the dashed lines have been
straightened out by pulling the row K to the
left in Step 2 of the numerical example above
at each of the N stages.
Step 3. Add -A(I,l) times the Kth row to row I (where
I =} K), starting with the element in column 2 of
row I. Again, store each elerr.ent in the colurrm
to the left of its original position. Steps 2
and 3 together have the effect of advancing the
dashed lines, one column to the right.
Step 4. Increment K by 1.
Step 5. If K exceeds N, stop. Otherwise, return to
s·tep 1 .
To illustrate, consider the previous example again.
Step 0. K = 1
Step 1. 4.0 1.0 0.0 1.0
4.0 3.0 1.0 0.0
1.0 1.0 1.0 0.0
Note that the Kth column of the Identity matrix
is available by implication only and is not
stored in any vector or matrix.
-39-
l
1
]
]
]
]
]
]
.]
]
400.4 Step ') Pivot element has the value 4. 0. Divide row '- .
1 by 4. 0.
0.25 0.0 0.25
4.0 3.0 1.0 0.0
1.0 1.0 1.0 0.0
Step 3. Add -4.0 tines row 1 to rO\·T 2 1 -1.0 times r0\·1
1 to rO\·J 3 .
I 0, 25 0.0 0.25
2.0 1.0 -1.0
0.75 1.0 -0.25
Step 4. K = 1 + 1 = 2
Step 5. 2 t 3; Return to Step l.
Step l. 0.25 0.0 0.25 0.0
2.0 1.0 -1.0 1.0
0.75 1.0 -0.25 0.0
Step 2. Pivot element has the value 2 . 0 . Divide
row 2 by 2. 0.
0.25 0.0 0.25 0.0
0.5 -0.5 0.5
0.75 1.0 -0.25 0.0
Step 3. Add -0 . 2 5 times row 2 to rO\v 1, -0. 7 5 times
row 2 to rovT 3 .
-0.125 0.375
0.5 -'0.5
0.625 0.125
-40-
-0.125
0. 5·
-0.375
1 400.4
] Step 4 . K = 2 + 1 = 3
1 Step 5. 3 :}- 3. Return to Step 1.
Step 1. -0.125 0.375 -0. '125 0.0
1 0.5 -0.5 0.5 0.0
1· 0.625 0.125 -0.375 1.0
Step 2. Pivot element has the value 0.625. Divide
.1 rmv 3 by 0. 6 25.
-0.125 0.375 -0.1251 0.0
"1 0.5 -0.5 0.5 0.0
l 0.2 -0.6 1.6
Step 3. Add -(-0.125) times row 3 to row 1, -0.5 times
l row 3 to rovl 2
0.4 -0.2 0. 2
l -0.6 0.8 -0.8
) I 0.2 -0.6 1.6
Step 4. K = 3 + 1 = 4
") .. · Step 5. 4 > 3. Stop .
l In the program section th~t follows, note especially
the manner in which the elements of the Kth column of
J the Identity matrix are assumed to be available. Further,
] note also the necessity of generating the elements of the
Nth column outside the DO-loops.
]
] -41-
]
]
] 400.4
] NNl = N-1
:l DO 4000 K = l,N
T = 1.0/A(K,l)
1 DO 8000 J = l,NMl
8000 A(K,J) = A(K,J+l)*T
j A(K,N) = T
1 DO 9000 I = l,N
IF (I.EQ.K) GO TO 9000
j . BIG= -A(I,l)
DO 8888 J = l,NMl
:j "
8888 A(I,J) = A(I,J+l) + BIG*A(K,J)
'•j ·~
A(I,N) = BIG*T
9000 CONTINUE
'1 "'
4000 CONTINUE
I B. The Problem of Vanishing Pivot Element
An important tacit assumption in the above develop-
.1 ment is that the pivot element never has the value zero
~1 and hence,division by the value of that element is legal.
This is equivalent to the assumption that no submatrix of
<j A in the upper left is singular. (If matrix A is itself
singular, there is obviously no solution to the problem.)
] If the assumption does not hold, as in SUBROUTINE
J DETMT (A,DA,N), here also row interchanges need to be
performed to overcome the difficulty. If the Identity
J -42-
J ]
]
]
1· ]
J ]
.. ] ..
]
J ]
]
]
.]
]
1
400.4
matrix is assumed to be available, the following is a
simple exa111ple for illustration of the procedure.
0.0 2.0 1.0 0.0
3.0 4.0
Pivot element has the value of 0.0.
Interchange rows as division by zero is i10t per-
mitted. Note tha~ the rows of the Identity matrix are
also to be interchanged.
!3.0
[o.o Step 1. K = 1
4. 0 f
2. 0 l 0. 0 1.0
1.0 0.0
Step 2. Pivot element has the value 3.0. Divide
Step 3 .
Step 4 .
Step 5.
Step 2.
row 1 by 3.0
Add
jl.O 1.33 I 10.0 2.0
-(0.0) times row
0.0 0.33
1.0 0.0
1 to rmv 2 ..
leaves the matrices unaffected.
K = 1 + 1 = 2
2 .} 2. Return to Step 2.
Pivot element has the value 2 . 0 .
row 2 by 2.0.
1.0 1.33 0.0 0.33
0.0 1.0 0.5 0.0
Step 3. Add -1.33 times row 2 to roH. 1.
1
1.0
0.0
0.0
1.0
-0.67 0.33
o.5 o~o
-43-
This
Divide
I 1
J ., 4
J j
400.4
Step 4. K = 2 + l = 3
Step 5. 3 > 2. Stop.
Next, attempt the solution without the availability
of the Identity matrix.
Step 0. K = l
Step l. 0.0 2.0 1.0
3.0 4.0 0.0
Rows of A will be interchanged, but sine~ the Kth
column of the Identity matrix is available only by
implication, the elements here cannot be interchanged.
Hence
Step 2 •
Step 3 .
Step 4.
Step 5.
Step l.
r 3. o [o.o
4.0
2.0
Pivot element has
l by 3. 0.
l. 33 0. 331 I
0.0 2.0 l Add -(0.0) times
l. 33 0.33
2.0 0.0
K = l + l = 2
2 } 2. Return to
11.33 0.33
I l 2. 0 0.0
1.0
0.0
the value 3 . 0 .
0.0
row l to row 2.
Step l
0.0
1.0
-44-
Divide row
400.4
]
l .
1 l
l
J
l
l ]
]
.]
J
J
Step 2. Pivot element has the value 2.0. Divide row
2 by 2.0.
1.33 0.33 0.0
0.0 0.5
Step 3. Add -1.33 times row 2 to row 1.
0.33 -0.67
0.0 0.5
Step 4. K = 2 + l = 3
Step 5. 3 > 2. Stop.
The answer here differs from the previous (correct)
answer in that the colunms have been interchanged. This
is a direct consequence of our failure to interchange
rows of the Identity matrix in the modified method. The
remedy is simple. Merely interchange the columns.
The numerical example above provides the necessary
background for the development of the generalized ap
proach which follows.
C. Row Interchanges and Corrective I1odifications
Assume that the rows of matrix A are interchanged.
Rovl interchange is one of the elementary tran-sformations
and is equivalent to premultiplying a matrix by the cor
responding elementary transformation matrix. Briefly,
TA = C where C is the resulting matrix after the inter
change. Hence C-l = A-lT-l and c-lT = A-l.
-45-
J 400.4
]
]
]
]
]
]
]
]
]
J
J
]
]
]
]
]
The product c- 1T implies that the same elementary
• . -1 • transformatlon T operates on C ~ but this tlme post-
multiplication by T is involved. The conclusion is that,
to get A-l fro~ C-l, the colu~~s of C-l have to be inter-
changed to match the interchange of rows of A.
Applying t1"1e principles to the 2 x 2 exarr,ple, '"'e see
that, in effect, given A, a matrix C was generated by
interchange of rows of A, and C-l was found. The answer
thus obtained needs, therefore, further modification
(interchange of columns of c- 1).
v.Ji th a 2 x 2 matrix, the number of interchanges is
limited to 1. Also, if the interchange is necessary, it
will be at stage K = 1. In the case of larger matrices,
the number may be as high as N-1 and the necessity may
arise at some or all stages from K = l to K = N-1.
In general, v.1e may assume that the original matrix
after the required number of row interchanges has been
modified during the process itself to a new ma·trix C
1 -1 and A- = C T T l1 H-1
T T 2 1
The last and first row interchanges are represented
by TM and T1
respectively. In the modification, however,
the first and last colunm interchanges are represented
-46-
.,..]
.. '
J ].
J·
J
.J
]
"1 ..J
J
J
J
J
J
400.4
by T and T1
respectively . Iv1
So, if a number of row interchanges was involved,
the corresponding column interchanges must be performed
in reverse order. Appendix 1 contains a more formal proof.
D. The Best Pivotal Element
In the discussion so far, row interchange was sug-
gested only if the pivotal element was identically zero.
The pivot element could be small in absolute value and
hence division by that element and later modifications
of other elements may lead to inaccuracies.
On the one hand, there is the need to check whether
the pivot element is zero. If so, on the other hand,
there is the need to search other rows (limited to rows
K through N of colunm K so as to preserve the action of
the previous modifications of the reduction process) for
a non-zero element.
Combining all of these, as in SUBROUTINE DE':L'HT (l-.,DA,
N), assume that a search will be made for the largest
absolute valued element in the column containing the
pivot element. Three possibilities arise as a result of
the search.
l. Every element in the column is zero. The matrix is
then singular and A-l does not exist. (Note that the sub-
program does not produce an error message in this in-
stance. A message will result from the system when the
-47-
J 400.4
J
'-1 "
J
J
J
]
]
J
J ]
J
J
J
J
J ]
J
machine attempts later to handle infinity and the pro-
gram will be aborted.)
2. The pivot element itself is the largest in absolute
value and hence, no exchange is necessary.
3. An exchange is necessary.
Vector NEXCH is a bookkeeping vector which takes
into account items 2 and 3 above. The elements of this
-1 vector get defined during the reduction process as C
is generated and are used later to perform the correspon-
ding column exchanges (in reverse order) .
E. The Determinant of the Original Matrix A as a By-
Product
A comparison of the procedures involved 1n this sub-
program and SUBROUTINE DETMT (A,DA,N) shows that the re-
duction processes in the two cases have much in common.
In the latter case, the process was limited to reducing
the given matrix to an upper triangular matrix. In this
subprogram (see the reduction process assuming the exis-
tence of the Identity matrix), this process has been ex-
tended so that the given matrix is diagonalized. This
extension which merely includes more elementary transfor-
mations does not alter the value of the determinant. The
determinant is hence the product of the pivotal elements
(before their reduction to 1.0).
-48-
..
]
]
]
]
]
l
l
l
l ]
]
l
J ]
J
J ]
J _]
400.4 F. ~he Special Case of a 1 x 1 Matrix
An inspection of the DO-loop parameters in the final
version of the subprogram indicates that, if N = 1, the
ra~ges of some DO-loops would be from 1 to zero. This
difficulty is avoided by branching out control early in
the subprogram.
2.6.3 Features
It has been shown that (even) in the case of a general
square matrix (unsymmetric) , the inversion process can be
carried out within the-space of the original matrix itself,
provided no submatrix in the upper left o£ the original
matrix is singular. To handle matrices of a general nature,
only an additional vector of N elements is required. Speed
is considerably enhanced as constants are not generated
during execution.
2.6.4 Limitations
Original matrix A is destroyed.
2.6.5 Additional Remarks
It is not essential that NEXCH be a vector of integer
elements in the calling program. The use of that vector
is local to the subprogram and the user needs only to be
aware that any such vector, if defined prior to the CALL
statement in the calling program, may be destroyed when
this subprogram is used.
-49-
J 400.4
j
J
J _j
J j
J
J
J
.J
2.7 SUBROUTINE MOVE (A,B,M,N)
2.7.1 Function
The matrix B is defined to be the same as the given
matrix A. Both the matrices are of the same size :H rO\vs
by N columns.
2.7.2 Development of the Suborogram
The subprogram listed in Appendix 2 follows directly
from the relationship BI ,J = AI ,J· 'l'he matrices are
treated as single-subscripted arrays in the subprogram.
2.7.3 Features
In some cases, a matrix referenced in the CALL state-
ment of a calling program gets destroyed in the subpro-
gram when using the routines of this package.
Assume that the original matrix is needed later in
the calling program. In such situations, the use of
this subprogram, prior to a CALL to the routine where
the matrix gets destroyed, assures the availability of
the original matrix by a different name. The example
below illustrates this procedure.
CALL MOVE(A,B,N,N) CALL DETHT(A,DA,N)
The original matrix A is destroyed by the second
CALL statement. However, by virtue of the first CALL
statement, matrix B is the same as the original matrix A
and can be operated on as if it were matrix A itself.
-so-
~7)
_j
j
j
J j
J
400.4
In the example, if the second CALL statement is CALL
DETNT(B,DA 1 N) 1 matrix B gets destroyed and the original
matrix A is available for further operations by its mvn
name.
2.8 SUBROUTINE MULT (A,B,C,L,M,N)
2.8.1 Function
The product of matrix A (size L rows by M columns)
and matrix B (size M rows by N columns) is made available
to the calling program as matrix C (size L rows by N
colunms) . C = AB
2.8.2 Development of the Subprogram
In algebraic terms, the elements of matrix Care
given by !-1
C = I: A I,J K=l I,K
and the following subprogram may, therefore, be written.
SUBaOUTINE MULT (A,B,C,L,M,N)
CO~WON/IYENGAR/ I,J,K 1 Y (11)
REAL A ( L , 111) , B (I•'l , N ) , C ( L , N )
DO 1000 I = l 1 L
DO 1000 J = l,N
C(I,J) = 0.0
GO 10 0 0 K = 1 I f>'l
1000 C(I,J) = C(I,J) + A(I,K)*B(K,J)
RETURN
END
-51-
400.4
] It is worth noting here that for a specific set of
values of I and J, C(I,J) is completely defined only
] after the innermost loop is satisfied. Hence, instead
1 9f referring to C(I,J) which is a dotilile-subscripted
variable, it is advantageous to refer to a scalar quan-
] tity, say, SUM when this loop is being executed. This
modification saves time of execution.
To illustrate, assume two matrices each of size
] 100 x 100 are multiplied. The modified version listed
in Appendix 2 saves two million address computations.
] The saving in time in a trial run was approximately 6
] seconds (23.262 seconds vs. 17.148 seconds).
2.8.3 Limitations
J In general, each of the matrices A, B and C will dif-
J fer in size. Hence, no attempt is made in this sub-
program to store the product matrix C in either matrix A
or matrix B.
The size of matrix C will match the size of either
matrix A or matrix B or both, if matrix B or matrix A or
j both are square. The use of SUBROUTINE PMULT(A,B,K,L,X)
or SUBROUTINE POSTM(A,B,K,L,X) should be considered under
j these conditions.
In any case, when SUBROUTINE r-mLT(A,B,C,L,H,N) is
J used, the product matrix C should be distinct from both
j the matrices A and B, although matrices A and B may
·] -52-
J
] 400.4
]
2.9
]
]
]
1
1
l ]
]
]
J
J
J
J ., I
themselves be identical.
SUBROUTINE OUTE (A,I,J,TITLE,TITEL) and ENTRY OUTF and
ENTRY OUTG
2.9.1 Function
Matrix A of size I rows by J columns is printed.
Printing begins on a new page. Rows and columns are
numbered. Labels (up to 20 characters) provided by the
user appear at the top of each page.
If J ~ 10, and at the same time, I< 25, the matrix
is printed on one page. If J > 10, the first 10 columns
of matrix A are printed until all the rows (25 or less
to a page) are exhausted. Then the second 10 columns
Cor less) of matrix A are printed until all the rows
(25 or less to a page) are exhausted. And so on.
The user has the choice of printing the elements in
E-, F- or G-FORMAT.
2.9.2 Development of the Subprogram
A. Matrix Partitioning
The limits of 10 columns to a page and 25· rmvs
(double-spaced) to a page have been chosen taking into
consideration the number of characters that can be
printed per line (135 excluding the carriage control
character in column 1), the number of lines (about 63)
-53-
400.4
] per page, and ease of readability.
Since J colu~1s are to be printed, the nurr~er of ver-
tical partitions of the matrix is given by JHAX belmv.
The use of integer arithmetic (FORTPJill} should be noted
here. Jj\iAX = ( J + 9 } I 10
If 1 :5. J :5. 10, Jf.·IAX = 1
If 11 :::_ J < 20, Jt-lAX = 2, and so on.
In a similar way, t~1e number of horizontal partitions
] of the matrix is given by
IHl\X = (I + 24) /25
Schematically, the partitions are as in Figure 2.
j (I and J are assumed to be 58 and 45 respectively.
Hence, IMAX and Jt-iAX have the values 3 and 5 respectively.}
---~· J Columns ------
1,1 1,2 1,3 1,4 {. 1,5
I l~ows ·--i -
2,1 2,2 2,3 2,4 '!,' 2,5
j ---·
J 3,1 3,2 3,3 3,4 ?! I 3,5
Figure 2
j The required nunilier of pages for the printout is
. given by the product of JMAX and I.Lv1AX.
-54-
J j
] 400.4
]
1
] )
]
]
]
]
.1
]
]
]
.l
J
J j
j
J
J
The choice is made to print the matrix in vertical
partitions. Hence, for the example in Figure 2, the sub
matrix marked 1,1 will be printed on the first page. This
is followed by printing of submatrix marked 2,1 and then
by the one marked 3,1.
The process is next repeated for submatrices marked
1,2 and 2,2 and 3,2 in that order. And so on.
The outermost DO loop hence ranges . from 1 to Jiv1AX.
This is followed by the next DO loop whose range is 1 to
IMAX.
Consider next the items on a specific page. Let the
submatrix being printed on the page have indices IRm'iT
and JtOL.
The table below gives the first and the last row and
colunm ·numbers (of matrix A) printed on the page.
First row
Last row
First column
Last column
B. FORHAT Control
2 5 ( IROvV) - 2 4
25 (IRm'l) or I, whichever is less
10 (JCOL) - 9
10 (JCOL) or J, whichever is less
'l'he form of output (E- ,F- or G-FORHAT) is controlled
as follows. The FO&~T for printing elements of the
matrix is generated in an integer array IV. If the CALL
statement is CALL OUTE (A,I,J,TITLE,TITEL) 1 the character
-55-
]
]
"]
]
J ]
J
.1
1
1
J
j
J
j
J
j
J
400.4
stored in IV (2) is E in a suitable part of that (com
puter) word, and hence the output is in E-FORM1Vr. Simi
larly, CALL OUTF( ... ) and CALL OUTG( ... ) produce output
in F- and G-FORMAT respectively.
The X preceding Fl2.5 in IV (2) is deliberately built
in to ensure at least one blank space between two nuwbers
in a row, when printing in F-FOR1'1AT. This limits the
number of characters (digits plus the negative sign, if
any) to the left of the decimal point to 6. If an ele
ment has 7 or more characters to the left of the decimal
point, an asterisk will be printed by the machine
(CDC 6400) at the beginning of the field, warning the
user that the most significant digit/s and/or the nega
tive sign, if any, are not printed. Briefly, the number
is too "large" for a suitable printout under this FORNAT
control. On the other hand, if the number is too small,
only zeroes may appear on the printout. Under these con
ditions, as well as under conditions where the magnitudes
of the elements being output are unknown or unpredictable,
the E-FORMAT should be preferred.
The E-FO~ffiT output is not as easily readable as the
F-FORMAT output but this is a minor inconvenience, parti
cularly after experience is gained in reading the nillliDers.
(~hen G-FORMAT is specified, five significant digits
appear on output in F-FORJ.~ffiT if the number lies in the
-56-
J j
]
1-] -_
]
j
j
]
j
l ]
j
j
j
j
j
J J
400.4
range 100000.0 > absolute value ;:::: 0.1. Otherwise, the
number is output in E-FORl·1AT. That is, the machine makes
the choice between F-FOP11AT and E-FORHAT, depending on
the magnitude of the nu!T'ber to be output. The decimal
points in G-FORMi\.T output may not line up vertically and~
in general, the output will be a mixture of E- and F
FO&~ffiT outputs. Some elegance is thereby lost. On the
other hand, once G-FOID1AT output is specified, concern
need not be wasted on losing the most significant digits
(5 digits in this subprogram).
C. Labels
The user may label the matrix being output by using
appropriate 11 Values 11 for the arguments TITLE and TITEL.
These 11 Values 11 are Hollerith character strings up to a
maximum of 10 each.
The labels will ~ppear at the top of each page. Further,
these labels are followed by the word CONTINUED in paren
theses on the second and succeeding pages of output to
indicate that parts of the matrix have already been output
in previous pages.
The size of the matrix also is indicated by a printout
directly after labelling. If no labels are to be used,
TITLE and TITEL must be matched with lH~ and lH~ respec
tively (~ denotes blank). Examples on the use of labels
-57-
I J
400.4
J are given in the User's Guide (Chapter 4).
j 2.9.3. Limitations
A matrix with a large number of columns and only a few
j rows (say less than 10), when output using this subprogram,
will use several sheets of paper. Consider, for example,
a matrix 8 rows by 25 columns. The output, in this case,
j is on 3 sheets. If, however, the transpose of the matrix
is printed, the output is on one sheet. Apart from
j economy of paper, a further advantage of the latter pro-
j cedure is that an overall view of the matrix is available
without having to turn pages.
j The user should weigh these advantages against the in-
convenience of having to transpose the matrix and reading
j the transposed matrix.
1
j
j
j
j
.1 -58-
1
J 400.4
1
1
1
j
j
j
j
j
J
j
j
J
2.10 SUBROUTINE PMULT(A,B,K,L,X)
2.10.1 Function
The square ma,trix B of size L rows by L columns is pre-
multiplied by a rectangular matrix A of size K rows by L
columns and the product is returned to the calling pro-
gram in the rectangular matrix A. A = AB
2.10.2. Development of the Subprogram
In SUBROUTINE f·1UL'l' (A,B,C,L,N,N) 1 it was remarked that
the three matrices A, B, and C are of different sizes
when dealing with general rectangular matrices and hence,
matrix C could not be stored in either matrix A or matrix
B. If, however, matrix B is square, it follows from
[A] • [B] = [C] K X L L X L K X L
that the product matrix C is of the saw~ size as the rec-
tangular matrix A, which premultiplies the square matrix
B. Advantage is taken of this feature to save memory
space by storing matrix C in matrix A.
Schematically, the order in which the elements of the
product matrix are generated (row by row) is shown in
Figure 3.
A D Product r ;;-
Figure 3
-59-
400.4 Consider row 1 of matrix A. This row is used in con-
junction with each column of matrix B to generate elements
] in row 1 of the product matrix. After this is done, row
1 of matrix A is never referenced again. It is for this
reason that row 1 of the product matrix may be stored in
row 1 of matrix A. (The argument is valid for -all the rows
in turn.)
However, throughout the process of generating the
elements in row 1 of the product matrix, the elements 1n
row 1 of the original matrix A must be available. Hence
the necessity of a vector X of size L elements for inter-
mediate storage of the generated product elements.
The procedure involves, therefore, the following steps
j '
for all I from 1 to K.
1. Generate elements in row I of the product matrix
l and store these in vector X.
2. Transfer the contents of vector X to row I of
matrix A.
2.10.3 Features
:1 Nemory requirements are less when this subprogram is
used. A vector X of L elements will suffice to perform
j the operations, whereas, previously a matrix of size K
rmvs by L columns was required (SUBROUTINE MULT
J (A,B,C,K,L,L)).
j See also Section 2~11.5.
-60-
j
J
j_
j-
j
j
j
~.-)
.J
j
J
J
J J
400.4 2.10.4 Limitations
Matrix B must be square.
The advantage of saving of memory space is partly off-
set by a loss in speed of execution, as additional opera-
tions (transfer of vector X to a row of matrix A for each
I, I = 1 to K) are necessary.
Original matrix A is destroyed.
2.10.5 Additional Remarks
Matrix A may be rectangular or square.
If matrix A is square, it should differ from matrix B
at least by name. 'l'hat is, the product of a square matrix
and itself (say, matrix A times matrix A) cannot be ob-
tained with the use of this subprogram. SUBROUTINE I·lULT
(A,A,PROD,K,K,K) of this package is to be used to obtain
such a product in matrix PROD which is distinct from
matrix A.
-61-
"j. ,.
~ i . 1
'j
~·1 .. ..
'1 ] •.··· ., '
~~··
.. J ., .
.. ;..
t)J~
400.4
2.11 SUBROUTINE POSTM(A,B,K,L,X)
2 .11. 1 Function
The square matrix A of size K rmvs by K columns is
P?Stmultiplied by the rectangular matrix B of size K rows
by L colu~1s and the product is returned to the calling
program in the rectangular matrix B. B = AB
2.11.2 Development of the Subprogram
The logic here is essentially the same as in SUBROUTINE
PNULT(A,B,K,L,X). The elements of the product matrix are
generated column by colunm.
is represented by Figure 4.
------'~
_____ ,..,.. -
A B
Schematically, this process
r =~
Product
Figure 4
The process involves the follmving steps for all J
from 1 to L.
1. Generate elements in column J of the product matrix
and store these in vector X which is of size K
elements.
2. Transfer the contents of vector X to column J of
matrix B.
-62-
J
]
]
1-]-_
J
J
J
J
J
J
J
J
J
J
J
J
J
]
400.4
2 .11. 3 Features
Hemory requirements are less when this subprogram is
used. A vector X of K elements will suffice to perform
the operations, whereas, previously a matrix of size K
rows by L columns v1as required. (SUBROUTINE HULT
(A,B,C,K,K,L).)
2.11.4 Limitations
Matrix A must be square.
Original matrix B is destroyed.
The advantage of saving of memory space is partly off
set by a loss in speed of execution, as additional opera
tions (transfer of vector X to a column of matrix B for
each J, J = 1 to L) are necessary.
2.11.5 Additional Remarks
Matrix B may be rectangular or square.
If matrix B is square, it should differ from matrix A
at least by name. That is, the product of a square matrix
and itself (say, matrix B times matrix B) cannot be ob
tained with the use of this subprogram. SUBROUTINE MULT
(B,B,PROD,K,K,K) of this package is to be used· to obtain
such a product in matrix PROD which is distinct from
matrix B.
In the following discussion concerning the three multi
plication routines described so far, assume that matrices
-6 3-
]
1 ]
j
J j
J
J
,J
j
j
t
J
1 j
J _j
J
400.4
A and B are square and differ at least by name. For com-
patibility in matrix multiplication, they are necessarily
of the same size, say K rmvs by K columns.
The user, in this case, has a choice of subprograms as
indicated in the table below, the choice being governed
by the criterion of saving one, none, or both the matrices.
To Save
Matrix A Matrix B Neither matrix Both matrices
Use CALL Statement
CALL POSTH(A,B,K,K,X) CALL PMULT(A,B,K,K,X) Either of the above tv;o CALL HULT(A,B,C,K,K,K)
-64-
]
~ J.
J
j
.l
j
J
J
J
400.4
2.12 SUBROUTINE RDCBC(A,M,N) SUBROUTINE riDCOLG(A,M,N) SUBROUTINE RDRBR(A,M,N) SUBROUTINE RDROWG(A,M,N)
2.12.1 Function·
Values are read in from data cards for the elements of
matrix A which is of size M rows by N columns.
2.12.2 Development of the Subprogram
A." The FORMAT Declaration
For convenience in punching values, a field width of
10 columns for each piece of data is prescribed. Thus, 8
values can be punched per card. In brief, the FORMAT
used is (8Fl0.0).
This choice of FOR..."-iAT requires that the deci!llal point
be punched in the field unless it is to be assumed (by
the machine) to be at the end of the field.
On the CDC6400 it is permissible to punch data in E-
FORMAT also under the same FORHAT control. Care must,
however, be exercised to see that the exponent is placed
at the end of a field.
B. Order in Assigning Values
The order in which the values are assigned to the
elements depends on the user's choice of one of the four
routines. The details are given in the User's Guide
(Chapter 4).
-65-
400.4
2.12.3 Limitations
The decimal point must be punched in the field, unless
it is to be assumed to be at the end of the field. The
maximum number of characters (digits plus the negative
sign, if any) in any data piece is limited to 10 if the
decimal point is not punched and to 9, otherwi-se.
J 2.12.4 Additional Remarks
If an element is to be assigned the value zero, the
corresponding data field may, at user's option, be left
blank. In this case, the machine actually assigns the
value -0.0 to the element.
SUBROUTINE RDCOLG(A,N,N) is the fastest of the four
routines since the transfer of data occurs, with its use,
J in the natural order of storage of array elements (in the
1 machine) without any interruptions. It is also the most
economical one in terms of field length requirements (see
J Section 3.4).
J
J
-66-
1
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]
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]
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J
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400.4
2.13 SUBROUTINE SC1'1UL(A,H,N,X)
2.13.1 Function
Elements of the given matrix A of size M rows by N
columns are modified so that the fr'esh elements are X
times the original ones.
2.13.2 Development of the Subprogram
Each element of th3original matrix is multiplied by
the scalar X and the result stored back in the element
itself. The matrix is treated as a single-subscripted
array in the subprogram.
2.13.3 Limitations
The original matrix A is destroyed.
-67-
] 400.4
] 2.14 SUBROUTINE SINV(A,DA,N)
2'.14. 1 Function
] The matrix A, which is symmetric (size N rows by N
J columns) and positive-definite, is inverted and the in-
verse returned to the calling program in matrix A itself.
] The determinant of the original matrix A is computed and
returned in DA.
] 2.14.2 Development of the Subprogram
A. Introduction
] In civil engineering applications, the matrices that
need to be inverted are, in general, symmetric or can be
J rendered symmetric; e.g., stiffness matrix, flexibility
matrix. These matrices are also positive-definite.
If the features of symrnetry and posi 'ci ve-de finite
J nature are utilized in a program, the inversion process
can be speeded up, the comparison being with SUBROUTINE
l j MINV(A,N,DET,NEXCH) which can invert matrices that may be
symmetric or unsymmetric.
J B. Procedure
J The symmetric matrix A can be expressed in the form
A = A.A. T where A. is a lower triangular matrix.
J 1
_..!
J -68-
J ]
J 400.4
]
]
J.
.l
.J
J
]
]
j
J
]
]
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J
J
J
J
J
The elements of A are given by {i\ppendix 1)
A I IJ
AI ,J
From A
J-1 = (AJIJ - 2::
K=l
2 ~ A ) 2
JIK
J-1 1
= CZ\I IJ - 2:: A I I K "A J ,J K=l
= 0
AA T -1 ( A T) -1
= A = I
.AJIK) for
for
( A -1) = (A-l)T(
I> J
I< J
A -1)
The inverse of a lower-triangular matrix is also lower
triangular. If ~ = /, -l, the elements of ~ are
(Appendix 1)
~I ,J = 0 for I< J
~I I I 1 =
TI,I I-1
~I ,J = - 1 2:: A I ,K AI
1I K=J ~ K,J for I> J
Briefly, then, the procedure involves 1) factorization
of A into A and AT 1 2) obtaining A -l and 3) forming the
-1 T -1 -1 product (A ) ( A ) to give A .
The algorithm below shows that these operations can be
accomplished within the space of the original·matrix it-
self.
By symmetry, matrix A can be considered fully defined
if all the elements on and above the diagonal are known.
The space below the diagonal is hence available to store
-69-
. . ' : .... ·-· .... _···.
J 400.4
]
)
]
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]
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J
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J
]
J
elements of matrix A. . Since A. is a lower triangular
matrix, all its elements above the diagonal have the
value 0.0 and these elements will therefore neither be
generated or stored.
Regarding the diagonal elements of matrix A. , it is
fortunate that, once A.I,I is generated, AI,I is no longer
required and hence, it is possible to store A.I I in place . I
of AI I for all I. I
In fact, it is advantageous to store the reciprocal of
A.I,I in place of AI,I' since the diagonal elements will
then be elements of matrix (t..-1) or of matrix (A.-l)T.
Refer to the schematic representation of this process in
Figure 5.
The order of generating elements of matrix A. is as
follows. A. ~ First, l,l is computed as A l,l" Next, the
other elements of column 1 are computed from
1 = A. (A1 I) us.:j_ng the symrnetry of matrix A 1,1 I
(Al,I = A1 , 1 ). For all J in the range (N-1).2 J> 2, the
diagonal element is computed first, and the other elements
of column J next. When J = N, only the diagonal element
needs to be computed. -1
The next step is to generate matrix A. . The diagonal
elements of matrix (A. -l) T, _which are the same as those of
matrix A.- 1 , are already available. A.-l is a lower trian-
gular matrix. Since matrix A. occupies the lower half of
the matrix space at this stage and matrix A is no longer
-70-
] 400~4
J
]
]
]
J
J
]
]
J
]
J
J
J
]
required; matrix (A-l)T instead of matrix A-l is stored
in the upper half of the matrix space. Refer again to
the schematic representation. If now ~ = ( A-l) T, the
elements of matrix ~' which is upper-triangular, are
~ J I I = 0
1 ~I I I = AI I
I
1 ~ = -
J ,I AI I I .
for J >I
I-1
(K~J A I I K ~ J I K) for I> J
T The off-diagonal elements of matrix (A-l) are generated
in the subprogram, column by column.
The product (A-l)T(A-l) is next required. The result-
ing matrix is A- 1 , and the matrix A is no longer required.
Matrix A-l can hence be stored in the lower half of the
matrix space.
For clarity, assume X = (A-l)T, y = (A-1) = xT
and Z = A-l = (A-l)T(A~l) = XY .
Then ~ rJ
zi,J = ~ XI KYK J = ~ XI KXJ K since Y = K=l , I K=l I ,
Expansion of z I, J yields (for .I .2: J)
+ ... + X X I,I J,I
X x· + ... +X X . I,I+l J,I+l I,~ J,~
-71-
+
] 400.4
]
]
]
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]
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.J
J
J
J
]
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J
J
J
J
J
For L>H 1 XL1H is zero, since X is an upper-triangular
matrix. Hence 1 for I> J 1 .
N
= L: XI KXJ K K=I I I
Assume that the elements Z (elements of matrix A-1
) I,J
are generated rov; by rmv. The index I therefore varies
from 1 to N, while index J varies from 1 to I for each I.
A study of the following three expressions shows that, in
this scheme, once zi,J is generated, XJ,I is no longer
required.
ZI J = XI X + X X + I ,I J,I I,I+l J,I+l
Hence, xJ,I may be overwritten by the generated value
of zi,J each time. The result is matrix A- 1 .
·Elements
of A
Elements T
of (A. - 1)
Figure 5·
-72-
Elements T
of (A. - 1)
-1 A
Elements
) 400.4
]
]
]-
] ·_
]
]
J
]
J
]
J
]
J
J
J
J
J ]
C. Built-in Subprogram
The computations require summations of the form IFIN
L: AI, KAJ, K on four different occasions. K=INIT
It is well-
known that subprograms are used in such situations to save
coding and reduce the number of machine instructions.
Since, however, such a subprogram may not have the same
degree of usefulness in other programs or subprograms, a
compromise is struck by incorporating this feature in the
subprogram itself in an indirect manner.
This specific area of calculations in ·the subprogram
is referenced whenever the required summations are to be
computed, through the use of unconditional GO TO state-
ments (analagous to CALL statements). 'l'he RETURN state-
ment of a regular subprogram is simulated by an assigned
GO TO statement, of which the control variable is INDEX.
This variable is assigned the suitable statement label
prior to each occasion when the area of the "built-in"
subprogram is referenced. The four variables I, J, INIT,
IFIN also need definition. The DO-loop indices themselves
are generally used to account for this feature.
D. The Determinant of the Original Matrix as a By-Product
Since A= ~~T, det(A) = det(~).det(~T). Noting that
a matrix and its transpose have the same determinant,
det(A) = det(~).det(~). Since~ is a lower-triangular
-73-
] 400.4
]
]
]
~j
]
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]
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J ,-]
] .
J
j
J
J
]
J
J
matrix, its determinant is merely the product of its dia-
gonal elements. The computation of det(A) is thus inci-
dental to the inversion process.
E. The Case of a 1 x 1 Matrix
If N = 1, many DO-loops in the routine would have their
ranges such as 2 to 1. The difficulty is avo{ded by test-
ing whether N equals 1 early in the routine and if so,
calculating det(A) and A-l directly.
2.14.3 Features
Although the inverse of a sy~~etric matrix is symmetric,
it is conceivable that a symmetric matrix inverted by the
use of SUBROUTINE MINV(A,N,DET,NEXCH) does not result in
a perfectly sy1nrnetric matrix due to machine round-off
errors.
This subprogram, however, utilizes the property of sym-
metry of matrix A and hence, guarantees the symmetric
nature of its inverse.
The algorithm of this subprogram utilizes only the
elements on and above the diagonal of the original matrix
A. Hence , only these need be defined when using this
subprogram. The inverse \vill be fully defined in any case.
-74-
400.4
2.14.4 Limitations
Matrix A must be symmetric and also positive-definite.
It is destroyed.in the subprogram. The necessity of
matrix A being positive-definite requires some elabora
tion. If det( A) = x , det(A) = x 2 • If det(A) and hence
x 2 are negative, xis imaginary. Since det(A)- = x is the
product of the diagonal elements of matrix A, this imp
lies that the number of imaginary diagonal elements of
matrix A is odd.
If det(A) is positive, but a submatrix of A has a
negative determinant, there will be an even number of
imaginary diagonal elements in matrix A .
If det(A) is zero, matrix A is singular and its
inverse has no definition.
All of which leads to the conclusion that, if matrix A
is positive-definite, there will be no need to deal with
imaginary numbers. This restriction has, therefore, been
imposed in developing this subprogram and, in view of the
introductory remarks, is a justifiable one.
However, if matrix A is not positive-definite (and non
singular), its inverse does exist and the additional prob-
lem is that of handling imaginary numbers. It is possible
to modify this subprogram without actually using complex
number routines. This has been done .in another version
of this subprogram. An even more general subprogram
-75-
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J
J
J
]
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J
]
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J
J
J ]
J
J
J
J
l
400.4
employing pivoting during factorization of matrix A has
also been attempted. Hmvever, these are obviously longer
and slower in execution. They are not included in this
package.
2.14.5 Additional Remarks
In solving civil engineering problems, where the inver
sion of either the flexibility matrix or the stiffness
matrix is involved, it is recommended that this subprogram
be used in preference to SUBROUTinE MINV(A,N,DET,NEXCH),
at least during the development stage of a program.
For, the use of this subprogram will prevent full exe
cution if the matrix is not positive-definite. Logical
errors (or punching errors) in the generation of the
matrix are thereby indicated.
It is possible to rewrite this subprogram using only
about half the matrix space (upper or lower triangle plus
the diagonal). In such a case, it is suspected that the
maximum use of the built-in subprogram may not be feasible.
Further, a partial definition of the inverse is of
questionable benefit.
-76-
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400.4
2.15 SUBROUTINE SOLVE(A,B,N,L,DET)
2.15.1 Function
,
A system of linear simultaneous equations is solved.
2.15.2 Development of the Subprogram
A. Procedure
The system solved is AX = B. The dimensions of the
matrices are as under.
r•latrix
Coefficient Matrix A
Right-Hand Side B
Size
N rows by N columns
N rows by L col unms
Only tie spaces of matrices A and B are used and the
solution matrix X is returned in matrix B.
Matrix A is reduced to an upper-triangular matrix in
essentially the same manner as in SUBROUTINE DET:t•IT (A, DA, N) .
The only major difference is that, instead of column ex
changes, row exchanges are made, if necessary, in this
subprogram. This variation is required in order to pre-
serve the correct order of the elements in the columns of
the solution matrix X.
The reduction process is equivalent to forming suitable
linear combinations of the original set of equations.
Representing these operations as a set of elementary row
transformations given by T, we have, TAX= 'rB. Hence,
matrix B also needs modification in the same manner.
-77-
J 400.4
]
]
]
]
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1
]
j
J
l ]
1
J
j
j
J
At the conclusion of the reduction process,
simultaneous
Al,l Al,2
A2,2
(Zeroes)
equations have the form
Al,N ·
A2,N
A A • I,I I,N
A A N-l,N-1 N-l,N
X ••• N-l,K
B_ .L,K
B ••• N-l,K
BN, K ' ''
the
Starting with the last rmv and assuming that a typical
column K of matrix X is being generated,
A X = B . Hence X = B /A N,N N,K N,K N,K N,K N,N
AN-l,N-lXN-l,K + AN-l,NXN,K = BN-l,K
-78-
J 400.4
1
1
J ·J· ' .
J ]
J ]
J
J
J
J
J
For 1 S. I ~ N-1,
Hence, N
X = (B - L: A X ) I A I,K I,K J=I+l I,J J,K I,I
It is interesting to note the order in which the
elements of a column of the solution matrix X are
generated. The last element is generated first and the
first generated last. The process is hence generally
referred to as back-substitution.
Since BI,K is not required after XI,K is generated, it
is possible to store matrix X in matrix B.
B. The Determinant of the Coefficient Matrix as a
By-Product
The generation of this quantity is incidental to the
reduction process. Since the original matrix A has been
(upper) triangulated, the determinant is given by the·
product of the diagonal elements of the reduced matrix A.
c. The Special Case of N = 1
In this case, some DO-loops would have the range 1 to
0. The difficulty is avoided by branching out early in the
subprogram to make the evident direct calculations to get
final results.
-79-
l ~ , at
l
1 tJ
"J· .. .;.:
J
1 --J
400.4
2.15.3 Limitations
The original matrices A and B are destroyed.
2.16 SUBROUTINE SQTR(A,N)
2.16.1 Function
The transpose of the square matrix A of size N rows by
N columns is returned to the calling program in matrix A
itself.
2.16.2 Development of the Subprogram
A general rectangular matrix A of size M rows by N col-
umns and its transpose, matrix B of size N rows by H
columns differ in dimensions. If, however, matrix A is
square, its transpose also has the same dimensions. It
is, therefore, possible to transpose a square matrix in
its own space.
If N = 1, the matrix and its transpose are identical
and control is merely returned to the calling program in
this case.
The transpose operation does not affect the values of
the diagonal elements. These are, therefore, never re-
ferenced in the subprogram.
-80-
J 400.4
]
1
]_
] --
]
]
]
J
J
J
J
J
J
J
J
J
J ]
1
f, . ~.-l K Q. " K\. r~----- L+ INCR
N
~~~~~' :,~ t-~---SG-
\\_ \ -
Figure 6.
The matrix is treated as a single-subscripted array.
In Figure 6, consider the element in row K of column 1.
Proceeding parallel to the main diagonal from this element,
the subsequent elements (of the matrix when treated as a
single-subscripted array) have subscripts in increraents
of N+l and the last element has the subscript N2-N*(K-l).
The elements of the lower triangle can thus be referenced
using two DO-loops, one with index K for elements in
column 1, and the other with index L (for a specific K)
for elements parallel to the main diagonal.
For a typical element with subscript L in the lower
triangle, the matching element in the upper triangle has
the subscript L+INCR, where INCR = (K-l)*(N-1). These
elements are exchanged.
2.16.3 Features
The matrix is transposed in its own space.
-81-
1
]
]
i]• !•' I
I I
]
]
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]
]
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J ·~ j
J
J
400.4
2.16.4 Limitations
The matrix must be square. The original matrix is
destroyed.
2.17 SUBROUTINE SUB(A,B,C,M,N)
2.17.1 Function
Matrix B is subtracted from matrix A and the result
returned to the calling program in matrix C. C = A-B.
Each matrix is of size H rows by N columns.
2.17.2 Development of the Subprogram
In algebraic terms, ci,J = AI,J-BI,J" This subprogram
is hence the same as SUBROUTINE ADD(A,B,C,~1,N), except for
the sign. The matrices are treated as single-stiliscripted
arrays in the subprogram.
2.17.3 Additional Remarks
The resultant matrix C can be stored in either of the
original matrices (say B). In this case, the original
matrix (matrix B) will be destroyed.
-82-
J 400.4
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2.18 SUBROUTINE TrlULT(A,B,C,L,rvl,N)
2.18.1 Function
The transpose of matrix A (A is of size L rows by M
columns) is post-multiplied by matrix B (size L ro':ls by
N colurrms) to give matrix C (size M ro1:1s by N columns).
2.18.2 Development of the Subprogram
In civil engineering applications, products of the
type ATB are required in computations quite often. Using
two of the subprograms in this package, the product may
be obtained as follows.
CALL TRANS(A,TEMP,L,M)
CALL MULT(TEMP,B,C,M,L,N)
Such an approach requires a temporary matrix TEMP of size
N rov1s by L colurrms. The transpose operation, as \vell as
the storage required for TEMP, may be sa~ed by a slight
modification of the multiplication routine.
Assume that AT= TEMP and [TEMP]· [B] =C. An element
of C is then defined by
L
CI J = ~ TEMP • BK,J I K=l I,K
= by virtue of AT = TEMP.
This formulation shows that matrix TEMP does not need
either generation or storage.
-83-
,] I
]
]
J IJ J .1
400.4
2.18.3 Limitations
The product matrix C should be distinct from matrices
A and B. However, matrices A and B may be identical.
2.19 SUBROUTINE TRANS(A,B,H,N)
2.19.1 Function
The transpose of matrix A (size N rm·1s by N columns)
is made available to the calling program as matrix B
(size N rows by M columns).
2.19.2 Development of the Subprogram
The subprogram utilizes the property BJ,I = AI,J"
2.19.3 Limitations
If matrix A is square, its transpose (matrix B) is also
square. However, if they do not differ by name, the re-
sulting matrix will· always be symmetric. Hence, the
matrices should be distinct from each other in every case.
2.19.4 Additional Remarks
SUBROUTINE SQTR(A,N) is more efficient than this sub-
program in terms of memory space if a square matrix is to
be transposed and the original matrix may be destroyed.
-84-
J ]
J ] .
] -.
]
J
J
J
J
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J
]
]
l
400.4
2.20 SUBROUTINE XABATC(A,B,C,L,M,X)
SUBROUTINE XABTA (A,B,L,N,X)
SUBROUTINE XABTC (A,B,C,L,M,N)
SUBROUTINE XATBAC(A,B,C,L,M,X)
SUBROUTINE XATBB (A,B,L,N,X)
2.20.1 Function
Products of matrices i) C = ABAT, ii) A= ABT,
iii) C = ABT, iv) C = ATBA and v) B = ATB are obtained.
2.20.2 Development of the Subprograms
The concepts used in the earlier multiplication rou-
tines from the bases for the development of this set also.
The dimensions of the matrices are as noted in the User's
Guide (Chapter 4) for the several routines.
2.20.3 Additional Remarks
i) The multiplication routines of this package [barring
SUBROUTHJE SCIYIUL (A,M,N ,X)] may be classified into three
groups as under:
Grou2 1
HULT (C =
THULT (C = XABTC (C =
Group
AB) Pl:·1ULT POSTf-1
ATB) XATBB ABT) XABTA
Group. 3
XABATC (C = ABAT) XATBAC (C = ATBA)
-85-
(A (B (B (A
2
= AB) = AB) = ATB) = ABT)
J: '·
-..,~
.. .J
-.,] . . ~~ ,,
J
j
1
J
J
J
J
400.4
a. Grouos 1 and 2 have the following features:
Product Ivlatrix
Dimensions
Matrix Destroyed
Vector X
Speed of Execution
Hemory Requirement (Arrays}
Group 1
Storage is in a matrix C which is distinct from both the original matrices A and B in every case (that is, even if one or both the original matrices match the product matrix in size) .
All the matrices are assumed to be rectangular
(some or all of them may, however, be square).
None
Vector X is not required.
Faster
More
Group 2
Storage is in one of the ma~rices A or B, as indicated in the above notations.
Matrix A must be square, if the product is stored in matrix B, and vice versa (both the matrices may, however, be square).
The original matrix that bears the name of the product matrix finally (or, in other words, the matrix that is not assumed to be square) is destroyed.
Vector X is of size NS elements if the square matrix is of size NS rows by NS columns.
Slower, because of the additional operation of replacement of a column or a row by vector X.
Less, since vector X will suffice instead of a matrix.
b. Group 3 has the following features.
Hatrix B must be square as well as symmetric. A vector X
of size NS elements, if matrix B is of size NS rows by NS
columns, is required. £-latrix A may be rectangular or square.
The product matrix C is distinct from both the matrices A
-86-
] 400.4
J and B (even if matrix A is square). Matrix C is square as
well as symmetric. The original matrices A and B are not
J destroyed in the subprograms.
] ii) Other Products. Products as indicated on the left Of
J the following table may be obtained by the set of CALL
statements on the right, provided both the matrices A and
] B are square (the original matrix B need not be symmetric
1n any of these cases). All the matrices are assumed to
] be of size N rows by N columns. Vector X is of size N
elements. ]
Product CALL Statements
] A = ATB CALL s QTR ("-I i'J) CALL Pf-1ULT (A I BIN IN I X)
a.
] B = ABT CALL SQTR(B,N)
CALL POSTM(A,B,N,N,X) b.
l A = ATBA CALL XATBB(A,B,N,N,X)
CALL POSTM(B,A,N,N,X) c.
d. B = ATBA CALL XATBB(A,B,N,N,X)
] CALL H1ULT(B,A,N,N,X)
e. A = ABAT CALL XABTA(B,A,N,N,X)
] f.
CALL PMULT(A,B,N,N,X)
B = ABAT CALL POSTM(A,B,N,N,X)
] CALL XABTA(B,A,N,N,X)
In items c and e above, the original matrix B is des-
] troyed, even though the final product is formed in matrix
] A. The original matrix A is not destroyed when the final
product is in matrix B.
] -87-
]
]
400.4
3. GENERAL HOTES
3.1 COMMON Block Labelled IYENGAR
~ A. Purpose
In normal usage, a COlvJ.HON block establishes the required '] ·.
correspondence amongst the variables in the routines sharing
the block. Values of the variables are thus tran.smi tted
from one routine to another.
The purpose of the block in this package is totally dif-
;] ferent. It merely serves as a means to economize on the
memory space requirements for the temporary variables used
in the several subprograms.
SUBROUTINE OUTE(A,I,J,TITLE,TI'rEL) with ENTRY OUTF and
ENTRY OUTG uses 14 temporary variables, and this is the maxi-
mum number required in any routine [1vi th the exception of
SUBROUTINE EV (A, S, N) 'ivhich 'dill be treated as a special case
later in this discussion]. Consider, next, SUBROUTINE SINV
(A,DA,N) which requires 9 temporary variables. Together,
these two routines would need 23 locations in memory for
these variables. .
Assume, now, the existence of a COH!"l0N block 14 words long,
the length corresponding to the maximum number of temporary
variables in any routine. Let each set of temporary vari-
ables in the two routines belong to this block. Then, 14
locations will suffice for the variables, whereas previously
23 were required. The advantage is even more pronounced
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400.4
if the principle is extended by specifying that the temporary
variables of each routine in the entire package belong to the
same COHHON block. The same 14 locations still suffice for
all the temporary variables, as against 131 which would have
been required without the use of the COHNON block. (To main-
tain the length of the block at 14 in each subprogram, a
filler array Y of variable size is used wherever necessary.)
Regarding the values associated with the variables, it is
inuuaterial whether these variables belong to the CO.HHON
block or not. This is because of the nature of the variables
themselves. They get defined in a subprogram before actual
use in the subprogram. The values they finally attain, at
the tirr.e of exit from the ·subprogram, are inconsequential to
any other subprogram since they get redefined before use in
that subprogram.
SUBROUTINE EV(A,S,N) actually uses more than 18 temporary
variables by name. The number is reduced to 18 through the
use of an EQUIVALENCE declaration. Of these, 14 are inc-
luded in the COJ>lHON block. The other four variables FN, nm,
THRI and THRF are not. Recall that ·this subprogram has an
ENTRY IEV staternent for improvement of eigenvalues and eigen-
vectors. Assume that SUBaOUTINE EV(A,S,N) has. been CALL-ed
in the calling program. Then, the four variables listed
have some values in the subprogram at the time of exit from
the subprogram. These values need to be preserved for a sub-
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400.4
sequent improvement procedure, if any. The calling program
may use the improvement procedure immediately (that is,
before CALL-ing other subprograms). There is no special
prob~em if it does. If not, assume that other subprograms
in this package will be used by the calling program before
the improvement procedure. In such a situation, these vari-
ables (\vhose values are not regenerated during the improve-
ment phase) will be.affected in values if they are included
in the COHHON block. Their values are preserved by excluding
them from the block.
Finally, therefore, 18 memory locations are required for
the temporary variables in the entire package.
B. COHHON Block Label
cor-li•ION blocks may or may not have labels (or numbers).
The name blank COI1MON applies if the block has no label (or
number) . Else, the block is a labelled (or numbered) COf.lMON
block.
If blank C0~~10N were used in this package, the user would
be unable to specify another independent blank COMMON block
of variables of his own choice in the main program and other
(user's) subprograms, since there can exist only one blank
CO~~ON block in a program.
Labelled (or numbered) COM1-10N blocks can be numerous, the
maximum number depending on the r:tachine used. (In CDC 6400 at
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400.4
Lehigh, the number of labelled and numbered C01·IHON blocks is
limited to 61). One requirement of these blocks is that each
label (or number) used in a program must be unique.
The originator of this package has chosen his last name
for the label in the justifiable hope that a general user
will not dream up this identical label for any of his label
led COMMON blocks. All the same, it is necessary to emphasize
that the label IYENGAR should be considered taboo by the user
for any of his labelled cm,'lNON blocks when using this package.
If the label is used, the variables listed by the user in the
block may get redefined with each CALL to a subprogram of
this package.
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400.4 3.2 USE OF VARIABLE (ADJUSTABLE) DIMENSIONS
3.2.1 Adopted Procedure
In this packa.ge, the dimensions of all arrays used in
the subprogra~sare adjustable dimensions. The motivation
for this approach is explained in what follows.
A primary rule in communication between the· calling
program and the called subprogram is that the row dimen
sions of all double-subscripted arrays which are arguments
must match exactly in both routines. This follows from
the fact that double-subscripted arrays are stored column
by column in the machine.
Consider SUBROUTINE MULT(li.,B,C,L,H,N) as an example for
discussion in the rest of this section, Sec. 3.2.
The dimensions of arrays A, B and C are specified in
the subprogram by the declaration
REAL A(L,M) ,B(M,N) ,C(L,N)
If, in the calling program, a matrix A (of size 10 rows
by 15 columns) is to be postmultiplied by a matrix B (of
size 15 rows by 20 columns) to give the product matrix C
(of size 10 rows by 20 columns), the proper declaration
in the calling program is
REAL A ( 1 0 I 15 ) I B ( 15 I 2 0 ) I c ( 10 I 2 0 )
and the corresponding CALL statement is
CALL MULT(A,B,C,l0,15,20)
In this case, the dimensions match automatically. Note
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400.4
that the array sizes chosen are exact. Under-dimensioning
will not work in any scheme, and over-dimensioning does not
permit suitable matching since double-subscripted arrays
are involved.
3.2.2 A Limitation of the Adopted Procedure
Occasionally, a matrix with the same name but of dif
ferent sizes needs to be used in a calling program. The
dimension declaration in such a case must obviously account
for at least the maximum size of the matrix. (Note that a
subscripted variable can be dimensioned only once in a
program.)
As an example, matrix A may vary in size, say, from 2
rows by 3 columns to 10 rows by 15 columns. Then the di
mension declaration is
REAL A(l0,15)
Assuming that there are other similar matrices, the dec
laration may be
REAL A ( 10 , 15 ) , B ( 15 , 2 0 ) , C ( 10 , 2 0 )
No difficulty is experienced if the problem to be solved
involves the maximum dimensions; e.g., CALL HULT(A,B,C,lO,
15,20). However, when solving problems using dimensions
less than the prescribed maximum dimensions, the sizes will
not match. For example,
CALL MULT(A,B,C,6,9,12)
implies that, in the subprogram, matrix A has dimensions
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400.4
6 rows by 9 columns, whereas in the calling program it has
been dimensioned for size 10 rows by 15 columns.
To rectify the situation, the subprograms themselves
need to be modified slightly as suggested in the following
subsection.
3.2.3 Suggestion for Mod~fication by the User
If array sizes are variable in the calling program, the
required subprograms may be modified on the following lines.
The dimension declaration for the arrays in the calling
program should correspond to (at least) the maximum size
of each array in the program.
As an example,
REAL A ( 10 , 15 ) , B ( 15 , 2 0 ) , C ( 1 0 , 2 0 )
For matching the arrays in the subprograms and specifying
the particular sizes to be operated on, the folloltling is a
suitable set of statements.
In the calling program,
CALL .i:1IULT (A,lO,B,l5,C,l0,6,9,12)
In the subprogram,
SUBROUTINE HULT (A,NRA,B,NRB,C,NRC,L,H,N)
REAL A (NRA,l),B (NRB I 1) I c (NRC I 1)
Recall that whenever variable dimensioning is used in
subprograms, the array names as well as the variables
which represent the dimensions must be formal parameters.
The last three parameters in the argument lists represent
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400.4
the sizes ·of the matrices to be operated on. The rest pre
scribe the array names and maximum row dimensions.
The modified form is obviously suitable even if the
sizes are not variable in the calling program. Hence, an
additional advantage of this procedure is that the user
may choose suitable maximum dimensions for matr·ices in a
program for solving problems of a particular type. The di
mension declaration, in such a case, needs no further at
tention from problem to problem. However, it is worthwhile
to remember that memory space is wasted when problems that
do not involve maximum dimensions are solved.
The disadvantages of this procedure, attributable to
the longer argument list, are these:
1. Mistakes in matching arguments ln CALL sta~ements
are more likely.
2. There is a loss of efficiency since addresses of
a larger number of arguments need to be passed
back and forth by the system.
In the authors' opinion, thedisadvantages outweigh the
advantages. Also, the additional ·type of problem covered
by the modified procedure is of infrequent occurrence.
3.2.4 A Special Procedure
A few users have violated the rule of matching dimen
sions and used the earlier version (FC~~PK) of this package
successfully without any modifications.
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400.4
The procedure is stated here only for the sake of comp-
leteness of this discussion on variable dimensions. It is
not recommended for general use as special care needs to
be exercised in programming.
Assuming that the required matrices are overdimensioned,
the matrices in this procedure are defined thrqugh the use
of relevant subprograms only. As an example,
REAL A ( 1 0 ' 15 ) I B ( 15 I 2 0 ) I c ( 10 ' 2 0 )
CALL RDRm'1G (A, 6, 9)
CALL RDROWG (B,9,12)
Here, 54 elements of matrix A and 108 elements of matrix
B get defined. The sequence of storage of the elements in
this case, although not unpredictable, is dependent on the
extent of over-dimensioning and will hence change \vi th each
problem .. For the sake of brevity, the elements may be said
to be in "wrong" locations. If, now, CALL HULT (A,B,C,6,
9,12) is used, the machine again refers to the same wrong
locations of elements in matrices A and B. Since this is,
however, a consistent process in the machine, matrix C will
have proper values but its elements are again stored in
some other wrong locations.
The importance of printing each array through the use
of other subprograms is, by now, evident. In these sub
programs, the machine again refers to the same \vrong loca
tions and the printed values will therefore be right.
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In short, this procedure will be successful if the main
program always CALL-s the relevant subprograms whenever
arrays are to be defined through reading in data from cards,
operated on, or printed.
Wastage of memory space can be considerable if the
special procedure is adopted in solving problems where the
sizes of the matrices in the calling program remain statio
nary.
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] 400.4
3. 3 RESERVED NAlvlES
When executing a FORTRAN main program with several sub-
programs, it is essential that each of these bear a unique
j name so that the machine does not get "confused". The rules
of operation vary with each (machine) system when this re-
j quirement is violated. In the following, the discussion
applies to the CDC 6400 installation at the Lehigh University
Computing Center.
] If more than one routine by the same name is loaded in
the machine, the one that was loaded first is recognized by
] the machine as the one meant for use~ Assume that this pack-
age is loaded first in the machine and the user's subprogram
] SUBROUTINE SOLVE (X,Y,Z) is loaded subsequently. Since the
] matrix package was loaded first and contains a subprogram
SUBROUTINE ·SOLVE (A,B,N,L,DET), the user's subprogram
] SUBROUTINE SOLVE (X,Y,Z) will be ignored during execution.
If this is the intention, there is no problem.
] The converse is equally true. If the user's subprogram
] is loaded first, the subprogram SUBROUTINE SOLVE (A,B,N,L,DET)
of the matrix package will be ignored.
] Hence, it is best to treat all the 30 names used for the
routines in this package as reserved names, and choose other
l names for the user's program and subprograms.
*For loading sequence, see User's Guide (Section 4.4).
l -98-
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l Further, it is obligatory that the name of every variable
j used in a routine differ from the name of any subprogram
CALL-ed by the routine. ,
]. (A renunder -- the name IYENGAR lS a reserved name when
labelling CO~L.'-'iON blocks.)
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3.4 SUBPROGRAH LENGTHS
The length of each subprogram in octal numbers, using the
RUN and FTN compil~rs of the CDC 6400 installation at Lehigh,
is indicated belmv. Small variations in these numbers are
likely with periodic revisions to the compilers. The length
of the COM£.10N block is 16 8 •
Sl. No.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Subprogram
ADD DETMT DIAG EV (IEV) GEVP MINV HOVE MULT OUTE (OUTF,OUTG) P~1ULT
POSTM RDCBC RDCOLG RDRBR RDROWG SCMUL SINV SOLVE SQTR SUB TMULT TRANS XABATC XABTA XABTC XATBAC XA'rBB
RUN
31 141
33 405 146 230
26 53
233 63 62 46 27 46 50 24
217 265
50 31 53 36
105 64 54.
102 62
CO~ll-'ION block 16 ---TOTAL 3555
Length (octal) FTN (OPT=2)
42 221
43 364 215 254
37 77
224 105 101
52 34 52 52 40
255 362
65 42
103 57
130 102
74 131 104
16
4436
[The earlier version (FCHXPK) required 4265 8 words (RUN)
and had fewer routines.]
-100-
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The package uses two routines, SQRT and ABS, available in
the machine.
The following notes are especially ,for users of the
CDC 6400 at Lehigh.
Although this package may be catalogued as one unit on a
permanent file, it is not necessary to load the entire pack-
age each time the file is attached. Selection of routines
is possible through the use of control cards (COPY routines)
such as COPYN, to minimize field length requirements for the
. b ( 2) JO • Note, however, that if SUBROUTINE GEVP (A,B,S,T,N)
is selected, other routines as mentioned in .the User's Guide
(Chapter 4) must also be selected.
SUBROUTINE OUTE (A,I,J,TITLE, TITEL) must be selected if
any of the three print routines (OUTE,OUTF,OUTG) is required
by a calling program.
Selection of SUBROUTINE EV (A,S,N) assures the avail-
ability of the associated ENTRY IEV.
Finally, such selection by the.user may become unnecessary
in due course, because it is anticipated that the loader it-
self will be modified to select routines, as required, for
each job. (J)
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3.5 TEMPORARY VECTORS
A few subprograms of this.package require the use of tern-
porary vectors to store intermediate values in computations.
The rnatching vectors in the calling program must obviously
be dimensioned. Such vectors, if they have been defined in
the calling program, will be destroyed in the subprograms.
It is sufficient to provide for only one temporary vector
in the calling program, even if it uses more than one sub-
program of the above type. The vector should be dimensioned
for the maximum size required in the use of all such subpro-
. grams. It may then be used repeatedly in all the correspond-
ing CALL statements. (Alternately, the several vectors that
result may be EQUIVALENCE-d at their starting addresses to
save memory space.)
SUBROUTINE GEVP(A,B,S,T,N), in this regard, is a special
case. It requires the use of a temporary matrix S in addi-
tion to that of a temporary vector T. In the calling program,
matrix S may be matched either by a matrix or by a vector of -
N*N elements. Assuming matrix S is matched by a vector, the
b1o vectors corresponding to S and T must be independent of
each other in the calling program. In other words, they do
not share the same memory locations.
-102-
400.4
] 3.6 Featur~s of th~ CDC ·6400
The routines in this package were developed using the
] CDC 6400 computer at Lehigh. Some of the features of this
machine are noted below, since minor modifications may be
necessary when this package is used in other machines.
] 1. The name of a variable, routine, label etc.·, is
limited to 7 characters. ]
In this package, only the C0Mi"10N block label (IYENGAR)
] is 7 characters long. All other names used are shorter.
2. On the line printer, a maximum of 135 characters (not
] counting the carriage control character in column 1) can be
] printed on a line.
Practically full advantage of this feature is taken
] in the output routines of this package.
3. An ENTRY statement has no associated argument list.
However, in the CALL statement corresponding to the ENTRY
] statement, the argument list (if any) of the routine in which
the ENTRY statement appears must be matched.
] Example: CALL IEV(A,S,N)
4. Conversion of an integer to its real form may be done
] by an assignment statement across the equality slgn.
.l See line 34 of the coding in SUBROUTINE EV(A,S,N)
5. The .length of each computer word is 60 bits. Hence,
J it is possible to store 10 alphanumeric .characters in each
word.
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400.4
A maximum of 20 characters may thus be used to match the
bw variables TITLE and TITEL in the output routines of this
package. The other entities in these routines using the same
featt;tre are arrays Y and IV, and the A-field in the FOFJ-'lAT
statement.
3.7 Use of Single Subscripts in Selected Routines
In only a few routines, the (double-subscripted) matrices
are treated as single-subscripted arrays. These are some of
the simpler routines which could be so written with the ad-
vantages of saving memory space and execution time. Trials
with a few other routines showed that such modifications lead
to one of the advantages at the expense of the other, the
execution time being more generally the item hurt. Further
probing in ·this area seems warranted.
-104-
400.4
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4. USER'S GUIDE
4.1 INTRODUCTION
The subprograms in this package were developed by Hr .
Sampath Iyengar of the Computer Systems Group in Fritz Engi-
neering Laboratory, for use by members of the Laboratory.
Dr. C. N. Kostem is the Chairman of the Computer Systems
Group.
This version supersedes the earlier one (FCMXPK - fortran
Callable Matrix Package) which will be wi thdra1'.·m by a date to
be specified. Mr. Edward T. Manning, Jr. was associated
with Hr. Iyengar in the development of FCMXPK.
In this Guide, only a brief description of the function,
limitations and requirements of each subprogram is included.
Nore detailed information on how the subprograms were de-
veloped are available in Chapters 2 and 3.
The package includes routines for matrix manipulation as
under:
OPERATION
Add matrices Determinant of matrix Create Diagonal matrix Eigenvalues of symmetric Invert matrix Copy matrix Hultiply matrices
Print matrix Read matrix
Simultaneous equations Transpose matrix Subtract matrices
SUBPROGRA.H
ADD DETHT~ HINV, SINV, SOLVE DIAG
matrices EV, IEV 1 GEVP MINV 1 SINV MOVE
-105-
HULrr, PNULT 1 POSTl'-1 1 SCHUL 1
THULT, XABATC, XABTA 1
XABTC, XATBAC, XATBB OUTE 1 OUTF, OUTG RDCBC 1 RDCOLG 1 RDRBR 1
RDROh1G SOLVE SQTR 1 TRANS SUB
.1
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400.4
4.2 GENERAL LIMITATIONS
The subprograms do not provide any diagnostics if opera-
tions which are not possible mathematically, such as inver
sion of a singular matrix, are attempted. The requirements
of the subprograms are not tested prior to or during execu-
tion and hence, when the requirements are violated, the
answers obtained are, in general, unpredictable and wrong.
Only ·the following subprograms provide printouts:
i) SUBROUTINE OUTE (A,I,J,TITLE,TITEL)
ii} SUBROUTINE OUTF (A,I,J,TITLE,TITEL)
iii} SUBROUTINE OUTG (A,I,J,TITLE,TITEL)
SUBROUTINE RDCBC (A,M,N), SUBROUTINE RDCOLG (A,M,N), SUB-
ROUTINE RDRBR (A,M,N) and SUBROUTINE RDROWG (A,M,N) enable
reading in values from data cards for the matrix A. All
others are "calculation" subprograms.
In all the subprograms, "variable" or "ad)ustable" dimen
sions are used for the several arrays. The user must, there-
fore, prescribe exact dimensions for all his arrays to be
handled by this package. Overdimensioning, except under
special circumstances of usage (Chapter 3), may lead to wrong
results. Underdimensioning invariably produces wrong results
or aborts the execution of the program.
For Input-Output operations, card input and printer output
are assumed.
The limitations on the sizes of the matrices that can be
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400.4
handled are not due to any feature in programming involved in
this package but due to the capacity of the machine used.
The label IYEI:JGAR may not be used in the user's program
for c;my of his labelled CO.N.HON blocks. Similarly, the names
of routines in this package may not be used in the user's pro-
gram for his program or any of the subprograms.
4.3 DISCLAIMER
The burden of proof on the validity and applicability of
this package to a particular problem rests with the user, and
not the authors. No guarantee is stated or implied that the
package will give correct results, or that the mathematical
relations and assumptions used are proper and applicable to
the problem under consideration by the user. The authors
cannot be held responsible for incorrect results or damages
resulting from the use of the package, although it is be
lieved that the package is correctly formulated.
The authors welcome suggestions for improvements and
notice of any errors. If a correction is possible and imp
lemented, proper pUblicity will be given to the revised
status of the package. Else, the concerned subprogram will
be withdrawn.
4.4 DECK SETUP
This package is expected to be available as a permanent
file at Lehigh University Computing Center, in due course.
-107-
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400.4
At the present tine, a prospective user who belongs to Fritz
Laboratory may borrow a binary deck and make his own copy.
Interested users will please contact the authors.
A sample deck setup, using the binary deck, is as follmvs:
Job Card ... RUN(S) LOAD(INPUT) LGO. 7/8/9 User's FORTRAN Program with Subprograms, if any 7/8/9 Binary Deck (FLMXPK) 7/8/9 Data, if any 6/7/8/9
The matrix package will be loaded first and the user's
program next. To reverse the order, replace thecontrol cards
LOAD(INPUT) and LGO. by LOAD(LGO) and INPUT., respectively.
4.5 DESCRIPTION OF THE SUBPROGRAMS
From the user's viewpoint, there are, in all, 30 routines
in this package. These will be described in alphabetical
order under the following headings:
a. Function:
b. Calling Program:
i) Dimensions: Those that are required in the calling
program.
If the calling program is the main program, the di-
mensions must be stated in terms of absolute nu~bers,
such as
-108-
~1 400.4 , REAL A (10, 15). If it is a subprogram, the dimension
statement is either of the same form or of the form
l - REAL A (H,N) v1here H and N have been defined through
:). .
operations prior to the use of the subprogram. (The
user is obliged, in the latter instance, to include
~-. A, M and N in the argument list of his su?program.)
All the subscripted variables handled by this
l .package are "real" variables. [The solitary excep-
l tion, vector NEXCH in SUBROUTINE MINV (A,N,DET,NEXCH) 1
needs no special consideration by the user.] Mistakes
~ often occur when this fact is overlooked and the user
prescribes an "integer" name for what is clearly an
l array of "real" variables. An example from civil
l engineering is to refer to the stiffness matrix as K
without a corresponding TYPE statement. An easy solu-
1 tion is to dimension the arrays in a TYPE statement
such as REAL K (20,20), TEMP (20). An advantage here
J is that the user has a free choice of names. Further,
J ..
~
a separate DIMENSION statement need not be (in fact,
should not be) provided. Inclusion in the TYPE
.J statement of names of arrays which are "real" even
without a TYPE statement, like TEMP in the example,
_] does not hurt in any way.
~ ii) Definitions: Arrays and variables that must be
defined prior to or in the CALL statement.
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400.4
iii) Values Returned to the Calling Program:
c. Limitations (if any) :
The general limitations mentioned in Section 4.2 will
not be repeated.
c. or d. Additional Notes (if any):
c. or d. or e. Examples of CALL Statement:
4.5.1 SUBROUTINE ADD (A,B,C,M,N)
a. Function:
Add matrices A and B and store the sum in matrix c.
(C = A + B) •
b. Calling Program:
i) Each matrix is of size M rov1s by N columns.
ii) Matrices A and B, as well as integers M and N, must
be defined.
iii) Hatrix C is defined in the subprogram.
c. Additional Notes:
The resultant matrix may be stored in one of the
original matrices. Only in such a case, the specific
original matrix will be destroyed.
d. Examples:
i) CALL ADD (A,B,C,M,N)
ii) CALL ADD (A,B,C,l5,20)
iii) CALL ADD (A,B,A,l0,15)
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400.2
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4.5.2 SUBROUTINE DETMT (A,DA,N)
a. Function:
The determinant of the given (square) matrix A is made
available to the calling program as DA. DA = det(A).
b. Calling Program:
i) Matrix A is of size N rows by N columns. -
ii) Matrix A and integer N should be defined.
iii) DA is defined in the subprogram.
c. Limitations:
The original matrix A is destroyed.
d. Examples :
i) CALL DETI•IT (A, DA, N)
ii) CALL DETHT (ARRAY,DET,20)
4.5.3 SUBROUTINE DIAG (A,DA,N)
a. Function:
A diagonal matrix A is generated as follows:
Each diagonal element has the value DA, and each off-
diagonal element has the value zero.
b. Calling Program:
i) Hatrix A is of size N rows by N columns.
ii) The value of the diagonal element DA and integer N
must be defined.
iii) Matrix A is defined in the subprogram.
' I . ...
-111-
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400.4
c. Exarnoles:
i) CALL DIAG (A, DA, N)
ii) CALL DIAG (,A 1 1. 0 1 15)
iii) CALL DIAG (ARRAY,DE,NSIZE)
4.5.4 SUBROUTINE EV (A,S,N)
a. Function:
Eigenvalues and eigenvectors of the symmetric matrix A
are computed.
b. Calling Program:
i) Hatrices A and S are of size N rows by N columns.
ii) Matrix A and integer N must be defined.
iii) On return to the calling program, matrix A has, for
its diagonal elements, the eigenvalues of the origi-
nal matrix A and matrix S has, for its columns, the
corresponding eigenvectors.·
c. Limitations:
The original rna trix A rnus t be symmetric. It ~.vi 11 be
destroyed in the subprogram, as the eigenvalues are re-
turned in the same matrix .
d. Additional Notes:
The eigenvalues and eigenvectors may be improved
further, if so desired, by using SUBROUTINE IEV (A,S,N).
e. Examples:
i) CALL EV (A,S,N)
ii) CALL EV (ARRAY,EVEC,lO)
-112-
"1 ijJ
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400.4
4.5.5 SUBROUTINE GEVP (A,B,S,T,N)
a. Function:
Eigenvalues and eigenvectors of the given matrix A
where [A] (X} =\[B] (X} are compu~ed. Both the matrices
A and B are symmetric, and further matrix B is also
positive-definite.
b. Calling Program:
i) Natrices A, B and S are of size N rows by N columns.
T is a vector of size N elements.
ii) Matrices A and B, as well as integer N, should be
defined.
iii) The eigenvalues are returned as the diagonal elements
of matrix A and the corresponding eigenvectors as the
colunms of matrix B. Hatrix S and vector T are used
fo~ storing some intermediate values in computations.
c. Limitations:
Matrices A and B must be symmetric. Matrix B must also
be positive-definite. Both the original matrices A and B
are destroyed in the subprogra~.
The following subprograms of this package must be
available and loaded when this subprogram is used:
i) SUBROUTINE EV (A,S,N)
ii) SUBROUTINE NULT (A,B,C,L,M,N)
iii) SUBROUTINE POSTM (A,B,K,L,X)
-113-
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400.4
d. Examples:
i) CALL GEVP (A,B,S,T,N)
ii) CALL GEVP (EVAL,EVEC,S,TEMP,lO)
4.5.6 SUBROUTINE IEV (A,S,N)
a. Function:
Eigenvalues and eigenvectors of the symmetric matrix A
computed by the use of SUBROUTINE EV (A,S,N) are improved.
b. Call~ng Program:
Same as in SUBROUTINE EV (A,S,N).
c. Lim~tations:
Same as in SUBROUTINE EV (A,S,N).
d. Additional Notes:
The accuracy of calculations in SUBROUTINE EV (A,S,N)
is prescribed according to the following scheme. The
square root of the sum of the squares of the elements
above the major diagonal (of the original matrix A) is
computed first. This is called the initial threshold. A
final threshold value of one-millionth of such sum is then
established. The diagonalization, which is an iterative
process, proceeds up to the stage when the absolute value
of every off-diagonal element is less than or equal to
the final threshold value.
Since the process is iterative, the user has the option
to improve the accuracy of the results by successive CALL-s
to the subprogram. For reasons explained in the
-114-
] 400.4
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documentation (Chapter 2), these successive CALL-s must be
to SUBROUTINE IEV (A,S,N). The following rules apply.
i) SUBROUTINE EV (A,S,N) must be CALL-ed once only and
before the SUBROUTINE IEV (A,S,N) is CALL-ed.
ii) SUBROUTINE IEV (A,S,N) may be CALL-ed subsequently the
required number of times to achieve the desired accu-
racy. If a total number of n CALL-s are made to (both)
the subprograms, each off-diagonal element will be
reduced in absolute value to (at least) 10- 6n times
the initial threshold.
iii) Neither matrix A nor matrix S may be altered in the
calling program between any tv10 of the above CALL-s
to the subprogram.
Trial runs have indicated that the improvement proce-
dure causes small but significant changes in the
eigenvectors, and practically no cha~ges in the eigen-
values (apparently because these are already very
close to the exact values). An excessive number of
improvement cycles may result in an underflmv in the
machine.
e. Example:
CALL EV (A,S,N)
CALL IEV (A,S,N)
CALL IEV (A,S,N)
-115-
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400.4
4.5.7 SUBROUTINE HINV (A,N,DET,NEXCH)
a. Function:
The matrix A is inverted in its 0\'m space and its
determinant is computed.
b. Calling Program:
i) Matrix A is of size N rows by N columns. Vector NEXCH
is of size N elements (see item d below).
ii) Matrix A and integer N should be defined.
iii) The inverse of the original matrix A is returned in A
itself. The value of the determinant of the original
matrix A is returned in DET.
c. Limitations:
The original matrix A is destroyed in the subprogram.
See also "Additional Notes" under SUBROUTINE SINV (A,DA,N).
d. Additional Notes:
The vector NEXCH is used for computations only in the
subprogram and the values of its elements are of no con-
sequence to the calling program. Hence , the matching
vector in the calling program need-not necessarily be a
vector of integer elements.
3. Examples:
i) CALL MINV (A,N,DET,NEXCH)
ii) CALL HINV (ARRAY,lO,DET,TEMP) ..
l
j.
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]
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400.4
4.5.8 SUBROUTINE MOVE (A,B,M,N)
a. Function:
Matrix A is copied as matrix B. (B = A)
b. Calling Program:
i) Matrices A and B are of size H rows by N columns.
ii) Hatrix A and integers M and N should be defined.
iii) Matrix B is defined in the subprogram.
c. Additional Notes:
When certain subprograms such as SUBROUTINE DETMT
(A,DA,N) of this package are CALL-ed, the original matrices
get destroyed in the subprograms. The user may have a need
to store the original matrices for further use at a later
time. This subprogram meets such a need.
d. Examples:
i) CALL MOVE (A,B,H,N)
ii) CALL NOVE (ARRAY,SAME,l0,15)
4.5.9 SUBROUTINE 11ULT (A,B,C,L,M,N)
a. Function:
Matrix A is post-multiplied by matrix B to yield matrix
C. (C = AB)
b. Calling Program:
i) Matrices A, Band C have the following dimensions:
Matrix Size --A L rows by M columns B M rov1s by N columns c L rows by N columns
-117-
] 400.4
] ii) Matrices A, B and integers L, M and N should be
defined.
iii) The product matrix C is defined in the subprogram.
J c .. Limitations:
Matrix C should be distinct from matrices A and B.
] However, matrices A and B may be identical. See also
SUBROUTINE PHULT (A,B,K,L,X) and SUBROUTINE POST11
(A I B 1 K 1 L 1 X) •
] d. Examples:
i) CALL MULT (A,B,C,L,M,N}
] ii) CALL MULT (A,A,C,N,N,N)
"]'· ..
The examples beloH yield \vrong results:
iii} CALL MULT (A,B,A,L,H,M}
J iv) CALL HULT (A,B,B,L,L,N)
v} CALL MULT (A,A,A,N,N,N)
] 4.5.10 SUBROUTINE OUTE (A,I,J,TITLE,TITEL}
J a. Function:
l'iatrix A of size I rows by J columns is printed. Print-
ing begins on a new page. The matrix is labelled at the
J top of each page with the labels provided by the user. The
word CONTINUED in parantheses appears against the label if
.J printing is on more than one page. The size of the matrix
is indicated below the label.
J Rows and columns are numbered. On any one page, the
] -118-
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400.4
maximum number of rows printed is 25, and the maximum
number of columns is 10. Hence, if J ~ 10 and I~ 25,
printing is completed on one page. If J > 10, the first
10 columns are printed, until all the rov1s (25 or less to
a page) are exhuasted. Then the second 10 columns (or less)
are printed, until all rows are exhuasted, and so on. The
elements of matrix A are output in E-FOm,mT. Five digits
appear to the right of the decimal point (El2.5).
b. Calling Program:
i) Matrix A is of size I rows by J columns.
ii) Matrix A and integers I and J should be defined. Also,
the user's label must be provided as a Hollerith
string of characters (maximum 20) through alphanumeric
variables or "values" corresponding to the arguments
TITLE and TITEL. See examples of CALL statement.
iii} No formal "values" are returned by this subprogram.
c. Additional Notes:
This subprogram is recommended for use in preference to
SUBROUTINE OUTF (A,I,J,TITLE,TITEL) whenever the magnitudes
of the elements of the matrix to be printed are unknown,
unpredictable, or exceed the field Fl2.5. A slight
sacrifice of easy readability is implicit.
d. Examples:
i) CALL OUTE(A,I,J,TITLE,TITEL)
-119-
l 400.4
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ii} VAR = 10 Hi\IATRIX OF TITLE = lOHREDUNDANTS. CALL OUTE(A,I,J,VAR,TITLE}
iii} CALL OUTE(A,I,J,lOHORIGINAL M,5BATRIX)
4.5.11 SUBROUTINE OUTF(A,I,J,TITLE,TITEL)
a. Function:
All. the details are the same as in SUBROUTINE OUTE(A,I,
J,TITLE,TITEL) except that the elements are output in F-
FORl.'1AT. Five digits appear to the right of the decimal
point, and a maximum of six digits (five, if the value is
negative} appear to its left. (Fl2.5}
b. Calling Program:
Same as in SUBROUTINE OUTE(A,I,J,TITLE,TITEL}
c. Limitations:
The "largest" numbers that can be printed are of the
form abcdef. ghijk or -bcdef. ghijk. If a number "larger"
than these is attempted to be printed, an asterisk (*) will
appear at the beginning of the corresponding field.
c. Examples:
i) CALL OUTF(A,I,J,TITLE,TITEL)
ii) CALL OUTF (A, I I J I 8Hr--<;.]J..TRIX AllH
4.5.12 SUBROUTINE OUTG(A 1 I,J,TITLE,TITEL}
a. Function:
All the details are the same as in SUBROUTINE OUTE(A 1 I,
J 1 TITLE,TITEL) 1 except that the elements are output in G
FORMAT.
-120-
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400.4 b. Calling Program:
Same as in SUBROUTINE OUTE(A,I,J,TITLE,TITEL)
c. Additional Notes:
Five significant digits appear on output, if the abso-
lute value x of the element being printed is in the range
0.1 ~ X < 10 5 (G 12.5}
Otherwise, the output is in E-FORMAT for the element.
d. Examples:
i} CALL OUTG(A,I,J,TITLE,TITEL}
ii) CALL OUTG(A,I,J,lH ,lH)
4.5.13 SUBROUTINE PMULT(A,B,K,L,X}
a. Function:
The square matrix B is premultiplied by a rectangular
(or square) matrix A and the product matrix is stored in A.
A = AB.
b. Calling Program:
i} Matrices A and B, and vector X, have the following
dimensions:
Hatrix Size
A K rows by L columns (may be square, K = L)
B L rows by L columns
X (Vector) L elements
ii} Matrices A, B and integers K, L should be defined.
-121-
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400.4
iii) The product matrix is returned in matrix A. Vector
X is required in the subprogram for computations only.
c. Limitations:
Matrix B must be square. The ori·ginal matrix A is
destroyed. If matrix A is square, it should differ from
matrix B at least by name.
d. Examples_:
i) CALL P.iYlULT (A,B,K,L,X)
ii) CALL PI>1ULT (A,B,K,K,X)
The example below yields wrong results:
iii) CALL PMULT(A,A,L,L,X)
4.5.14 SUBROUTINE POSrrr-I(A,B,K,L,X)
a. Function:
The square matrix A is postmultiplied by a rectangular
(or square) matrix B and the product matrix is stored in B.
B = AB.
b. Calling Program:
i) r1atrices A and B, and vector X, have the following
dimensions:
Matrix
A
B
X (Vector)
Size
K rows by K columns
K rows by L columns (may be square, K = L)
K elements
ii) Matrices A, B and integers K, L should be defined.
-122-
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iii) The product matrix is returned in matrix B. Vector
X is required in the subprogram for computations
only.
c. Limitations:
Matrix A must be square. The original matrix B is
destroyed. If matrix B is square, it should differ from
matrix A at least by name.
d. Examples:
i) CALL POSTH(A,B,K,L,X}
ii} CALL POSTM(A,B,L,L,X}
The example belmv yields wrong results:
iii} CALL POSTM(A,A,K,K,X}
4.5.15 SUBROUTINE RDCBC(A,H,N}
a. Function:
Elements of matrix A are defined (column by column} by
reading in values from data cards.
b. Calling Program:
i} Matrix A is of size H rows by N columns.
ii} Integers M and N should be defined.
The number of data cards required per column of
matrix A is (M+7}/8. (Integer division}
iii) Matrix A is defined in the subprogram .
c. Additional Notes:
i} FORHAT Control:
The FOR}ffiT is (8Fl0.0} and the decimal point should
-123-
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400.4 .I ~
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preferably be punched in each data field. If it is
not punched, it will be assumed to be at the end of
the field. The non-punch positions in the field are
assumed to be filled with zeroes.
For example, if 23.5 is punched beginning in column
21, the value assigned to the corresponding element
is the same. If the decimal point is not punched,
the value assigned is 2305000000.0.
ii) Order in Assigning Values:
Assume matrix A is of size 14 rows by 6 columns.
Two data cards are required per column. Hence, the
total number of data cards required is 12. The
eight values on the first data card will be assigned
in order to Al,l' A2 , 1 , .•. ,A8 , 1 • The six values on
the second data card to A9 , 1 , AlO,l''''' Al4,1"
And so on.
d. Examples:
i) CAI.L RDCBC (A, M, N)
ii) CALL RDCBC(A,l4,6)
4.5.16 SUBROUTINE RDCOLG(A,H,N)
a. Function:
Elements of matrix A are defined by reading in values
from data cards. The elements are assumed to be in a con-
tinuous string of columns of matrix A.
-124-
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400.4
b. Calling Program:
i) Hatrix A is of size H rows by N columns.
ii) Integers M and N should be defined.
The number of data cards required is (M*N+7)/8.
(Integer division}
iii) Matrix A is defined in the subprogram.
c. Additional Notes:
i) Same as in SUBROUTINE RDCBC(A,M,N}.
ii) Order in Assigning Values:
Assume matrix A is of size 14 rows by 6 colunms.
Eleven data cards are required. The eight values
on the first data card will be assigned in order to
Al,l' A2 , 1 , ... ,A8 , 1 . The first six values on the
second card to A9 , 1 ,A10 , 1 , ... , A14 , 1 . The last two
values on the second card to A1
,2
, A2
,2
. The eight
values on the third card to A3 , 2 , A4 , 2 , ... , A10 , 2 .
And so on.
iii) Of the four READ subprograms in this package, this
one requires minimum exe~ution time.
d. Examples:
i) CALL RDCOLG(A,H,N)
ii) CALL RDCOLG(A,l4,6)
4.5.17 SUBROUTINE RDRBR(A,M,N)
a. Function:
Elements of matrix A are defined (rmv by row) by
-125-
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400.4
reading in values from data cards.
b. Calling Program:
i) lv:latrix A is of size H rows by N columns.
ii) Integers M and N should be defined.
The number of data cards required per row of matrix
A is (N+7)/8. (Integer division}
iii) Matrix A is defined in the subprogram.
c. Additional Notes:
i) Same as in SUBROUTINE RDCBC(A,M,N)
ii} Order in Assigning Values:
Assume matrix A is of size 14 rows by 6 columns. One
data card is required per row. Hence, the total num-
ber of data cards required is 14. The six values on
card number I (I ranges in value from 1 to 14) are
assigned in order to AI,l' AI, 2 , ... , AI,6"
d. Examples:
i) CALL RDRBR(A,M,N)
ii) CALL RDRBR(A,l4,6)
4.5.18 SUBROUTINE RDROWG(A,M,N)
a. Function:
Elements of matrix A are defined by reading in values
from data cards. The elements are assumed to be in a con-
tinuous string of rows of matrix A.
b. Calling Program:
Same as in SUBROUTINE RDCOLG(A,M,N)
-126-
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400.4 c. Additional Notes:
i) Same as in SUBROUTINE P~CBC(A,M,N) •
ii) Order in Assignina Values:
Assume matrix A is of size 14 rm·1s by 6 columns.
Eleven data cards are required. The first six values
on the first data card will be assigned in order to
Al,l' A1 , 2 , ... , A1 , 6 . The last two values on the
·first card to A2 , 1 , A2 , 2 . The first four values on
the second card to A2 , 3 , A2 , 4 , A2 , 5 , A2 , 6 . The last
four values on the second card to A3 , 1 , A3, 2 , A3,3'
And so on.
d. Examples:
i) CALL RDROWG(A,M,N)
ii) CALL RDROHG(A,l4,6)
4.5.19 SUBROUTINE SCMUL(A,M,N,X)
a. Function:
Elements of matrix A are multiplied by the scalar
quantity X.
b. Calling Program:
i) Hatrix A is of size H rows by N columns.
ii) Matrix A and the scalar multiplier X as well as
integers M and N should be defined.
iii) The modified matrix is returned in A itself.
c. Limitations:
The original matrix is destroyed.
-127-
400.4
d. Examples:
i) CALL SCMUL(A,M,N,X)_
ii) REI = 1. 0/EI
CALL SCMUL(A,H,N,REI)
iii) CALL SCMUL(A,M,N,l.0/30000.0) '
J 4.5.20 SUBROUTINE SINV(A,DA,N)
a. Function:
The symmetric matrix A (also positive-definite) is in-
l verted in its mvn space and its determinant is computed.
b. Calling Program:
i) Hatrix A is of size N rows by N columns.
ii) Matrix A and integer N should be defined. The sub-
program utilizes only the elements on and above the
l diagonal of the original matrix A. Hence, if so
desired, only these elements of matrix A may be
defined.
1 iii} The inverse is returned in matrix A itself. DA
stores the value of the determinant of the original
matrix A.
c. Limitations:
The original matrix A must be symmetric as well as
] positive-definite. The original matrix is destroyed.
d. Additional Notes:
The inverse of a syiTmetric matrix is also symmetric.
This property has been utilized in this subprogram, and
-128-
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400.4
hence the resulting inverse of a symmetric matrix will be
symmetric when this subprogram is used.
It is possible that the inverse of a symmetric matrix ob-
tained by the use of SUBROUTINE MINV(A,N,DET,NEXCH) is
not ideally symmetric because of round-off errors in the
machine.
e. Examples:
i) CALL SINV(A,DA,N)
ii) CALL SINV(K,DETK,20)
4.5.21 SUBROUTINE SOLVE(A,B,N,L,DET)
a. Function:
A system of linear simultaneous equations &~ = B is
solved.
b. Calling Program:
i) Matrices A and B have the following dimensions:
Hatrix
Coefficient Matrix A Right-Hand Side Matrix B
Size
N rows by N columns N rows by L columns
ii) Matrices A and B as well as integers N and L should
be defined.
iii) The solution matrix is returned in B. The value of
the determinant of the coefficient matrix A is
returned in DET.
c. Limitations:
The original matrices A and B are destroyed.
-129-
']-. •.'
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d. Examples:
i) CALL SOLVE(A,B,N,L,DET)
ii) CALL SOLVE(COEFF,RHS,N,1,DA)
4.5.22 SUBROUTINE SQTR(A,N)
a. Function:
The transpose of the square matrix A is returned to the
calling program in A itself.
b. Calling Program:
i) Matrix A is of size N rows by N columns.
ii) Matrix A and integer N should be defined.
iii) The transposed matrix is returned in_ A itself.
c. Limitations:
Matrix A must be square. The original matrix is
destroyed.
d. Examples:
i) CALL SQTR(A,N) ·
ii) CALL SQTR(ASQ,lO)
4.5.23 SUBROUTINE SUB(A,B,C,M,N)
a. Function:
Subtract matrix B from matrix A and store the result
in matrix c. (C = A -B)
b. Calling Program:
i) Matrices A, B and C are each of size M rows by N
columns.
-130-
·-1 . ,~·.' 400.4
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ii) Matrices A and B, as well as integers M and N should
be defined.
iii) Matrix C is defined in the subprogram.
c. Additional Notes:
The resultant matrix may be stored in one of the ori-
ginal matrices. Only in such a case, the spe_cific origi-
nal matrix will be destroyed.
d. Examples:
i) CALL SUB(A,B,C,M,N)
ii) CALL SUB(A,B,C,lO,lS)
iii) CALL SUB(A,B,B,25,30)
4.5.24 SUBROUTINE TMULT{A,B,C,L,M,N)
a. Function:
The transpose of matrix A is postmultiplied by matrix
B to give matrix C.
b. Calling Program:
i) Matrices A, Band C have the following dimensions:
Matrix
A B c
Size
L rmvs by H columns L rows by N columns N rows by N columns
ii) Matrices A and B, as well as integers L, .H and N,
should be defined.
iii) The product matrix is returned in matrix c.
c. Limitations:
Matrix C should be distinct from matrices A and B.
-131-
400.4
However, matrices A and B may be identical.
d. Examples:
i) CALL TNULT(A,B,C,L,H,N)
ii) CALL TMULT(A,A,B,L,M,M)
iii) CALL TMULT(A,A,B,N,N,N)
The examples below yield wrong results-:
iv) CALL TMULT(A,B,A,L,L,L)
v) CALL TMULT(A,B,B,M,M,H)
j vi) CALL TMULT(A,A,A,N,N,N)
4.5.25 SUBROUTINE TRA.J.~S(A,B,H,N)
a. Function:
j Matrix A is transposed to give matrix B. (B -- AT)
b. Calling Program:
J i) Matrix A is of size ry1 rows by N columns. Matrix B is of size N rows by H columns.
.I ii) Matrix A and integers M and N should be defined.
iii) Matrix B is defined in the subprogram.
c. Limitations:
Even if matrix A is square-, it should be dis tinct
from the (square) matrix B.
J d. Additional Notes:
....
J If matrix A is square and the original matrix may be
destroyed, SUBaOUTINE SQTR(A,N) should preferably be used.
] e. Examples:
i) CALL TRP~S(A,B,M,N)
J -132-
1
J 400.4
] . ii) CALL TRANS(A,B,l0,15)
iii) CALL TRANS(A,B,N,N)
] The example belmv yields wrong results:
J .. iv) CALL TRANS(A,A,N,N)
4.5.26 SUBROUTINE XABATC(A,B,C,L,M,X)
] . . .
a. Function:
] The matrix product ABAT is formed in matrix C.
Matrix B is symmetric. C = ABAT
] b. Calling Program:
J i) .Hatrices A, B, C and vector X have the following
dimensions:
] Matrix Size --A L rows by M columns
J B M rows by H columns c L rows by L columns
X(vector) H elements
ii) Matrices A, B and integers L, M should be defined.
iii) The product matrix (symmetric) is returned in matrix
] C. Vector X is required in the subprogram for com-
] putations only.
c. Limitations:
] Matrix B must be symmetric.
Matrix C should be distinct from matrices A and B.
] d. Examples:
J i) CALL XABATC(A,B,C,L,M,X)
ii) CALL XABATC(A,B,C,L,L,X)
1 -133-
J
]
]
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]
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J 1 ,;1
]
1
l ]
]
1
400.4
The examples below yield wrong results:
iii) CALL XABATC(A,B,B,L,L,X)
iv) CALL XABATC(A,B,A,L,L,X)
v) CALL XABATC(A,A,A,L,L,X)
4.5.27 SUBROUTIHE XABTA(A,B,L,N,X)
a. Function:
The matrix product ABT 1s formed in matrix A. Natrix
B is square. A= ABT.
b. Calling Program:
~Matrices A, Band vector X have the following dimensions:
Matrix:
A B
X(Vector)
Size
L rows by N columns N rmvs by N columns N elements
ii) Matrices A, B and integers L, N should be defined.
iii) The product matrix is returned in matrix A. Vector X
is required in the subprogram for computations only.
c. Limitations:
.Hatrix B must be square. The original matrices A and B
should be distinct from each other even if matrix A is
square. The original matrix A is destroyed.
d. Examples:
i) CALL XABTA(A,B,L,N,X)
ii) CALL XABTA(A,B,N,N,X)
The example belov1 yields wrong results:
iii) CALL XABTA(A,A,N,N,X)
-134-
]
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]. ,.,., J
]
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J
400.4
4. 5. 28 SUBROUTINE XABTC (A ,B, C, L·,M, N)
a. Function:
The matrix product ABT is formed in matrix C. C = ABT
b. Calling Program:
i) Matrices A, B and C have the following dimensions:
!-latrix
A B c
Size
L rows by M columns N rov1s by .r-1 columns L rows by N columns
ii) Matrices A, B and integers L, M and N should be
defined.
iii) The product matrix C is defined in the subprogram.
c. Limitations:
Matrix C should be distinct from matrices A and B.
d. Examples:
'i) CALL XABTC(A,B,C,L,M,N)
ii) CALL XABTC(A,A,C,L,M,L)
iii) CALL XABTC(A,A,C,L,L,L)
The examples below yield wrong results:
iv) CALL XABTC(A,B,A,L,M,M)
v) CALL XABTC(A,B,B,L,L,L)
vi) CALL XABTC(A,A,A,L,L,L)
4.5.29 SUBROUTINE XATBAC(A,B,C,L,M,X)
a. Function:
The matrix product ATBA is formed in matrix C. Matrix
B is symmetric. C = ATBA
-135-
.,
.1 400.4
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]
]
]
]
1 ]
]
]
]
]
]
)
]
]
b. Calling Program:
i) Hatrices A, B and C and vector X have the following
dimensions:
Matrix
A B c
X(Vector)
Size
L rows by L rows by M rows by L elements
H columns L columns H columns
ii) Matrices A, B and integers L, M should be defined.
iii) The product matrix C (symmetric) is defined in the
subprogram. Vector X is required in the subprogram
for computations only.
c. Limitations:
!•latrix B must be symmetric. Matrix C should be distinct
from matrices A and B.
d. Examples:
i) CALL XATBAC(A,B,C,L,M,X)
ii) CALL XATBAC(A,B,C,L,L,X)
The examples below yield wrong results:
iii) CALL XATBAC(A,B,B,L,L,X)
iv) CALL XATBAC(A,B,A,L,L,X)
v) CALL XATBAC(A,A,A,L,L,X)
4.5.30 SUBROUTINE XATBB(A,B,L,N,X)
a. Function:
The matrix product ATB is formed in matrix B. Matrix
A is square. B = ATB.
-136-
400.4 b. ~allihg Program:
i) Hatrices A, B and vector X have the follmving
dimensions:
Matrix Size
A L rows by L columns
-,.. j .
B L rows by N columns X(Vector) L elements
~' . ii) Matrices A, B and integers L, N should be defined.
iii) The product matrix is returned in matrix B. Vector
X is required in the subprogram for computations
only.
c. Limitations:
Matrix A must be square. The original matrix B is
destroyed. The original matrices A and B should be
] distinct from each other even if matrix B is square.
d. Examples:
i) CALL XATBB(A,B,L,N,X)
ii) CALL XATBB(A,B,N,N,X)
The example below yields 1.vrong results:
iii) CALL XATBB(A,A,N,N,X)
... :4.
-137-
] ! '·
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]
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]
]
400.4
5. READY REFERENCE SHEET
After a few problems have been solved with the use
of this package, the user will have gained enough
experience to operate within its requirements and
limitations, and frequent references to the text
become unnecessary in day-to-day usage. All the.
same, matching the argument list of each subprogram
and dimensioning arrays suitably are problems which
may require some memory aid. A ready reference
sheet has, therefore, been provided. The sub-
programs are listed alphabetically.
-138-
\ -
.~ .~
NO. SUBPROGRAM 1. ADD(A,B,C,M,N) 2. DETHT(A,DA,N) 3. DIAG(A,DA,N)
4. EV(A,S,N) 5. GEVP(A,B,S,T,N) 6. IEV(A,S,N)
7. MINV(A,N,DET,NEXCH) 8. MOVE(A,B,M,N) 9. MULT(A,B,C,L,M,N)
10. OUTE(A,I,J,TITLE,TITEL) 11. OUTF(A,I,J,TITLE,TITEL) 12. OUTG(A,I,J,TITLE,TITEL)
13. PMULT(A,B,K,L,X) 14. POSTM(A,B,K,L,X)
I 15. RDCBC(A,M,N)
....... w 16. RDCOLG(A,M,N) 1..0 I 17. RDRBR(A,M,N)
18. RDROWG(A,M,N)
19. SCMUL(A,M,N,X) 20. SINV(A,DA,N) 21. SOLVE(A,B,N,L,DET)
22. SQTR(A,N) 23. SUB(A,B,C,M,N) 24. THULT(A,B,C,L,M,N)
25. TRANS(A,B,M,N) 26. XABATC(A,B,C,L,M,X) 27. XABTA(A,B,L,N,X)
28. XABTC(A,B,C,L,M,N) 29. XATBAC(A,B,C,L,M,X) 30. XA'l'BB (A I B' LIN, X)
.~ .L....... .~ .W .L-J
READY REFERENCE SHEET (FLMXPK)
BRIEF DESCRIPTION DIMENSIONS C = A + B A(H,N) ,B (M,N) ,C(M,N) Determinant A(N,N) Diagonal matrix 1\(N,N)
E-values,E-vectors A(N,N) ,S (N,N) Solve AX = A.BX A(N,N) ,B(N,N) ,S(N,N) ,T(N) Improve on EV A(N,N) ,S(N,N)
Invert A A(N,N) ,NEXCH(N) B = A A(M,N) ,B(M,N) C = AB A (L,M) ,B (M,N) ,C (L,N)
Print A A(I,J) Print A A(I,J) Print A A(I,J)
A = AB A (K,L) ,B (L,L) ,X (L) B = AB A(K,K) ,B (K,L) ,X (K) Read col. by col. A(M,N)
Read columns (grouped) A(M,N) Read row by row A(M,N) Read rows (grouped) A(M,N)
A = xA A(M,N) Syrn. inversion A(N,N) Sim. Eqs. (AX = B) A(N,N) ,B(N,L)
Transpose in place A(N,N) c = A. - B A(M,N) ,B (M,N) ,C (H,N) c = ATB A(L,M) ,B(L,N) ,C(H,N)
B = AT A(H,N) ,B(N,M) c = ABAT A(L,H) ,B(M,M) ,C(L,L) ,X(M) A = AB'l' A(L,N) ,B(N,N) ,X(N)
c = ABT A(L,M) ,B(N,M) ,C(L,N) c = ATBA A (L,M) ,B (L,L) ,C (M,M) ,X (L) B = ATB A (L,L) ,B (L,N) ,X (L)
. f
.~ ! _.._._.. _;:___J
NOTES
-.:--of:>. 0 0
400.4
1
APPENDIX 1
Proofs of Some Basic Theorems
1
-140-
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] -.
]
]
]
l
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J J J
400.4 'APPENDIX 1
In support of the relations which were used earlier in
the mathematical derivations, proof of some basic theorems
in matrix algebra have been included in this Appendix for
ready reference. For the sake of brevity, the proofs are
. given in only a few lines. A standard text, such as Refe-
renee 4, may be consulted for more detailed information.
The algebraic derivation of sin 9 and cos 9 from tan 29 is
also included.
1. The transp6se of a product is the product of the
transposes in reverse order.
Extending the proof,
2. The inverse of A exists iff (if and only if) the de-
terminant of A is non-zero.
Some authors define and others show that A-l =
1 det(A) [co-factor] where det(A) = determinant of A.
If det(A) ~ 0, the Right-Hand Side of this equation exists.
Hence, A- 1 exists.
-141-
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J
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"! ,;)
]
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]
.]
400.4
If A-l exists, the Left-Hand Side exists and, hence,
det(A) =I 0.
3. A-1 is unique.
If B and C are both inverses of A,
AB(=BA)=I and AC(=CA)=I,
and hence A(B-C) = 0, the null matrix.
Premultiply by B or c.
B - C = 0
Hence, B = C.
4. The determinant of the product is the product of the
determinants.
Consider 2 x 2 matrices.
I Al·l B I = Al,l AlI 2
A2,1 A2,2
This is the same as A A 1,1 1,2
0 0
A2,1 A2,2 0 0
-1 0 Bl,l B 1,2
0 -1 B . 2,1 B2,2
since, by Laplace expansion, using the first and second
rows, the result is
Add B times first column to the third, B2
,2
times 1,1
second column to the fourth, and then add B1 , 2 times
-142-
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first column to the fourth, B2
,1
times second column to
the third. The result is
Al,l Al,2 Al,1Bl,l+Al,2B2,1 Al,1Bl,2+Al,2B2,2
A2,1 A2 2 '
A2,1B1,1+A2,2B2,1 A2,1B1,2+A2,2B2,2
-1 0 0 0
0 -1 0 0
Again, by Laplace expansion, using the third and the
fourth rows, this is equal to IABj.
The proof may intuitively be extended to matrices
larger than 2 x 2.
5. The determinant of the inverse is the· reciprocal of
the determinant of the original matrix.
-1 1 AA =I, hence det(A)·det(A-) = det(I) = 1. This
also s~ows that, if det(A) = 0, det(A-1) = w, or the
elements of A- 1 are undefined.
6. The inverse of a product is the product of the
inverses in reverse order.
By definition, (AB) (AB)-l = I
Premultiply by A-l, B(AB)-1 = A- 1
Premultiply by B-1 , (AB)- 1 = B-1A-1
Extending the proof,
-143-
] 400.4
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]
7. The in:ve·rse ·of the transpose· is the transpose of the
inverse.
By definition, AA-l = I
By transpose, (A-l)TAT = IT = I
Hence, (A-l)T = (AT)-1
8. The inverse of a symmetric matrix is symmetric
(A-l)T = (AT)-1
Since AT = A, (A-l)T = A-1
9. The product of symmetric matrices is symmetric iff the
matrices commute with each other.
By definition, A and B commute if AB = BA.
(AB) T = BTAT
By symmetry of A and B, (AB)T = BA
If A and B commute, (AB)T = AB
Also (only if part),
if AB =(AB)T, AB = BTAT = BA or A and B commute
10. The product of a matrix and its transpose is
symmetric.
Note that A need not be square, and in general,
11. The product RT is orthonormal, if R and T are
othonormal.
-144-I
·~ 00. 4
i
Since T is orthonormal, RT (RT) T = RRT
Since R is orthonormal, RT (RT) T = I
Hence, (RT)T = (RT) -l
12. A symmetric matrix A may be expressed in the form
A =
AIIJ
A.A.T where A is lower triangular.
For I < J, AI,J = 0
For I 2:. J,
N
= L: A I KAJ K = K=l I I
Find elereents of A •
+ AI,JAJ,J + AI1J+l AJ,J+l +.·.+A I,N AJ,N
Since A. = 0 if L< M, L,M
J-1 A = L: AI K A +/.. A I,J K=l I J,K IIJ J,J
I-1 2 % If I = J, A = (A L:
·AI,K ) .
I,I I,I K=l
1 J-1 If I> J I AI,J = A . - L: AI, K A. J,K)
AJ,J I,J K=l
13. -1
A is lower-triangular if A is lower-triangular. Find
eiements of A- 1 .
By definition, AB = I vlhere B = A-1 . N
L: A B = 8I,J K=l I,K K,J (Kronecker Delta)
•.• +
-145-
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l
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J
Since
If I
lL = 0 for L < M, --L,M
I
5I,J = K~lAI,KBK,J
= 1 , and J > I , 6 · = 0 I,J
0 = A1 , 1B1 ,J , hence, Bl,J = 0
If I = 2 , and J > I , 6 I , J = 0
0 = A2 ,lBl,J + A2 , 2B2 ,J = 0 + A2 , 2B2 ,J, hence, B2 ,J = O
Similarly, for all J >I, BI ,J = 0
Hence, B = A-l is lower-triangular.
Non-zero elements of B.
If I = J, 6 = 1 I ,J
I 1 = L: AI,KBK,I =
K=l since both A and B are
lower-triangular. Hence, BI,I = --1-AI,I
If I>J, 5I,J = 0
BL,M = 0 if L< H. Hence,
0 = AI,JBJ,J + .•• + AI,I-lBI-l,J + AI,IBI,J
Hence, BI,J 1 I-1
= - -- ( L: AI KBK J) AI,I K=J ' ' .
for I> J.
-146-
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J
J
400.4 14. Given· tan 29 ,· find ·sin· ·9 and co·s a.
The value of tan 29 is given by tan 29 =
This problem arises in SUBROUTINE EV(A,S,N).
Let A = Numerator and ~ = Denominator.
Since tan a= tan ( a- n ) , there are two solutions for 28 in
the range - n to n • Since the solutions are supplements of
each other, a unique solution is available if the range is
chosen as - n
2 to n
2 The range of 9 is then -
n 4
n to 4·
sin 29 tan 29 = cos 29 = 2 sin 9 Vl - sin2e 1 - 2 sin2 9
By squaring, rearrangement and the substitution x = sin2 e,
2 A 2 A2 A 2 4x (1 + --) - 4x (1 + -) + -- = 0
~ 2 ~2 ~ 2
= 0.5[1 +} A2 X 2 2]
A +~
Since - ~ < 9 ,:s ~ , sin 2
9 or x has a maximum value of 1/2.
0.5(1- jl A 2. o.srA2;p.:2 +f.l.2)
~2J Hence, X = 2 2) = ll +) 1 ~,.z;o,2 A + ~ A2 +
If w2 = + w A2 + 2' sine = -~
j2 (l + )1 - w2) The signs of sine and tan 29 in the range chosen for 9 are
the same. The sign of tan 29 is governed by the signs of A
and ~ . The proper value of sin9 will therefore be available
with the definitions,
-147-
400.4
1
1
1
J
]
J
J
J
J
J
]
J
J
J
;\. w w = (sign of · 1-L ) -------, sine =
j ;\.2 +!J-2
The value of cose in the range is positive.
The advantages of this formulation are these:
1. sine and cose may be computed algebraically.
2. Even if !J. = 0 (and hence tan 2e is infinity), the
computations for sine and cose can be carried out
without difficulty in the computer.
.. 15. When the Identity Matrix is available by implication
only, the column exchanges, to match previous row exchanges
during reduction, must be performed.in reverse order. [See
SUBROUTINE t.fiNV (A IN I DET I NEXCH) ] •
With the availability of the I-t-latrix, tl:lis problem does
not arise as rows of I can also be interchanged during the
process itself.
When the !-matrix is not available, the Kth column of I is
available by implication only. When rows of A are inter-
changed, the net effect on I is that the corresponding rows
and columns of I are interchanged. Thus, if x1 is the trans
formation matrix corresponding to the rm·T interchange, we
have
-148-
400.4
at an intermediate stage of the reduction process and
at the final stage.
In short, I I -11 I I c '
Thus, SA = I or
] and A- 1x x ···x = c- 1 1 2 p
or A-1= c-1 -1 -1 -1 X • ··X X -1? 2 1
]
]
]
J '
] -149-
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':J• ~4
!o;P.:] <,;
-.~.
··~.~ .. ,
"··
'] ~
"' d
APPENDIX 2
Listing of Subprograms
-159-
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1
FRITZ LABORATORY MATRIX PACKAGE CFLMXPK) ------------------------------- PAGE NO, 1 -------------------------------
SUBROUTINE ADO (A,a,c,:·bN) 1 c C AUTHOR - SAMPATH IYENGAR, FRITZ ENGIN~ERING LABORATORY, C L~HIGH UNIVERSITY, BETHLEHEM, PA, 18015. c C THIS GROUP OF SU8P~OGRA~S CFLMXPK - FRITZ LABOR~TORY M~TRIX C PACKAGE> CONTAINS 30 ROUTINES TO PERFORM MATRIX OPERATIONS, C FOR DETAILED OOCUM~NTATION, SEE FLMXPK - A MATRIX PACKAGE, C FRITZ LABORATO~Y RE~ORT NO, 400,4. c
c
COMMON /IYENGAR/ I,MN,YC12> REAL A C 1) , 8 C 1) , C C 1>
C ROUTINE TO ADO MATRICES A AND 8. C=A+B c
c
MN=M.~;~.N
00 1GOO I=1,MN 1000 CCI>=ACil+BCI>
RF:TU RN END
SUB~OUTINE OETMT CA,OA,N>
C ROUTINE TO FIND THE DETERMINANT OA OF MATRIX A, c
c
c
COMMON /IYENGAR/ I,IP1,J,M,NM1,T,TEMP,XX,YC6)
0!\=1.0 IF OJ.EQ.t> GO TO 1050 m11= N-1 DO 1040 I=1,NM1 IPi=I+i
C SEA°CH qow I FQq THE LA~GEST ABSOLUTE-VALUED C ELEMENT IN COLUMNS I TH~OUGH N. c
c
TEMP=O.O 00 1000 J=I,N T=AI3SCA(I,J)) IF CTEMP.GE.T> GO TO 1000 TEMP=T
C THE LARGEST ABSOLUTE-VALUED ELEMENT IS IN COLUM~ M OF ROW I. c
M=J 1000 CONTINUE
IF CTEMP,NE.O.Ol GO TO 1010 c C ALL THE ELEMENTS OF ROW I ARE IDENTICALLY ZERO. C DETERMINANT VANISHES. c
c
OA=O.O RETURN
C EXCHANGE COLUMNS M AND I, ONLY IF M IS DIFFERENT FROM I. c
1010 IF CM.EQ.I) GO TO 1030
2 3 4 5 6 7 8 g
10 11 12 13 14 15 15 17 1~
19 20
1 2 3 4 5 5 7 9 9
10 11 12 13 14 15 15 17 1·3 19 20 21 22 23 24 25 26 27 2.13 29 30 31 32 33 34 35 35 37
I
FRITZ LABO~ATORY MATRIX PACKAGE CFLMXPK> ------------------------------- PAGE NO. 2 -------------------------------
c
DO 1020 J=I,N T=ACJ,MI ACJ,M>=ACJ,I>
1020 A(J,J>=T
C CHANGE THE SIGN OF TH~ DETERMINANT, C SINCE COLUMNS HAVE BEEN EXCHANGED. c
DA=-DA 1030 TEMP=A<I,I>
c C CONTINUOUS PART-P~OOUCT FOR THE DETERMINANT. c
DA=OA,..TEMP c C THE PROCESS OF REDUCTION TO AN UPPE~-TR.IANGULAR MATRIX. c
c
TEt~P=1. 0 /TEMP DO 1040 M=IP1,N' XX=- A CM, I> •TEMP 00 1040 J=IPi,N
1040 A(M,J)=A(M,J)+XX•Aci,J)
C FINAL VALUE OF THE JETERMINANT. c
1050 DA=DA,..A<N,N) RETURN EtW
33 3g 40 41 42 43 44 45 46 b7 48 49 so 51 52 53 54 55 5~
57 58 59 60 61 62 63 64-65
---------------------------------------------------------------------------SUBROUTINE QIAG C~1 DA,N>
c C ROUTINE TO CREATE A DIAGONAL MATRIX A. c
c
COMMON /IYENGAR/ I,NP1,NSQ,Y(11) REAL A<U
C ZERO THE ARRAY. c
NSQ=N'"N DO 1000 I=1,NSQ
1000 ACI>=O.O c C ASSIGN OA TO EACH DIAGONAL ELEMENT. c
N~1=N+1 00 1010 I=1,NSQ,NP1
1010 A<I>=DA RETURN END
1 2 3 4 5 6 7
g 10. 11 12 1"3 14 15
-15 17
• 1·g •1 g 20
-------------------------------------------------------------------------·--
l
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FRITZ LABORATORY MATRIX PACKAGE CFLMXPK> ------------------------------- PAGE NO. 3 -------------------------------
SUBROUTIN~ Ey tA,S,N> 1 c c c c
2 ~OUTINE TO FIND EIGENVALUES AND EIGENVECTORS 3 OF THE SYMMETRIC MATRIX 4. 4
5 CO~MON /IYENGAR/ AI 0 ,AIQ,APP 1 AQQ,COSSQ,COST,I,P,Q,QM1,SINSQ,SINT,T 6
1Et--IP, TSCT 7 EQUIVALENCE CAI::>,SIP,Y), CAIQ,SIQ), CP,IP1), (Q,J), CQ"H,NM1>, CTE 8
1MP,SIGN>, {THRI,SUM), <TSCT ,X) 9 LOGICAL INO 10 REAL ACN,N>,SCN,Nl 11 INT~GER P,Q,QM1 i~
c IF CN.EQ.1> GO TO 1010
c C CALCULATE TH~I, THE INITIAL THPESHOLD VALUE. C INITIALIZE MATRIX S AS THE IDENTITY MATRIX. c
c c c
c
SUM::O.O mlt=N-1 DO 1000 I::1,NM1 SCI,I>=t.O IP1::I+1 DO 1000 J::IP1,N T E i1 P :: A (! , J > SCI,Jl::O.b SCJ,I>-=0.0
1000 SUM=SUM+TEMP•TEMP TWU ::SQR T C SUM)
SET THE DESIRED ACCURACY <THRF>.
TH~F=THRI~1.0E-6 FN::N
C INITIALIZE INDICATOR IND. c
INO::. TRUE. 1010 SCN,N)::1.0
c ENTRY IEV
c C SUBROUTINE IEV CA,S,N> C ROUTINE TO IMPROVE EIGENVALUES AND EIGENVECTORS. c
IF CN.EQ.1> RETU~N IF CTHRI.EQ.O.O) RETURN
c C LOWER THE INITIAL THRESHOLD THRI, C SUCCESSIVELY BY DIVISION BY N. c
1020 THRI=THRI/FN c C ITERATION BEGINS HE~E. c
1030 DO 1050 Q=2,N QM1::Q-1 00 1050 P=1,QH1
13 14 15 16 17 1g 19 20 ? '
23 24 25 26 27 2<5 29 30 31 32 33 3 L;
3? 35 37 33 39 40 41 42 43 44 45 46 47 4:1 4'3 50 51 52 53 54
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FRITZ L~BORATORY MATRIX PACKAGE CFLMXPK) ------------------------------- PAGE NO. 4 -------------------------------· c 59 C TEST EACH OFF-DIAGONAL ELEMENT IN TURN. ~Q
c 51
c c c c
c c c
c c c c
1040 c c
·c
1050 c c c
c c c
X=-A CP,Ql IF <ABSCX) .LT.THRI> GO TO 1050
THE OFF-DIAGONAL ELEMENT IS NOT LESS THAN THRI IN ABSOLUTF VALUE. ANNIHILATE IT. RESET INDICATOR, SINCE A~NIHILATION IS N~C~SS~~v.
IND=.FALSE.
CALCULATE TRIGONOMETRIC FUNCTIONS.
APP=A CP,P) AQQ=ACQ,Ql Y=0.5""(APP-AQQ) SIGN=1.0 IF <Y.LT.O.O> SIGN=-1.0 TEMP=SI!,N~X/SQRTCX~X+Y~Y)
SINT=TEMP/SQRTC2.0~(1.0+SQRTC1.0-TEMP~TEHP)))
S Ii'-!SQ=SiiH"'SINT COSS 0=1. 0-SINSll COST=SQRTCCOSSQ) TSCT=-2.o~x~srNT~cosT
MODIFY ELEMENTS OF MATRICES A AND S ROW BY ROW. ALSO, TRANSPOSE TO MODIFY CORRESPONDING ROWS OF A.
DO 1040 I=1,N AIP=ACI,P> A IQ= A CI, Q)
TEMP=AIP~COST-AIQ~SINT
A(!, P> =TEMP A CP,!) =TEMP TEHP=AIP~SINT+AIQ~COST
A<I,Q>=TC::MP 1\(Q,I>=TEHP SIP=S<I,P> SIQ=SCI,Q> sci,P>=SIP~cosT-sra~siNT S(I,Q)=SIP""SINT+SI1~COST
EVALUATE FRESH ELEMENTS OF THE PIVOTAL SET.
ACP,P>=A0P~COSSQ+AQ1~S!NSQ-TSCT ACQ,Q)=AQQ•COSSQ+AP 0 •SINSQ+TSCT ACP,Q)=O.O ACQ,P>=O.O CONTINUE
CHECK WHETHER A REPEAT SHEEP IS NECESSARY.
IF ( INO) GO TO 1060
A REPEAT SHEEP IS NECESSARY.
IND=.TRUE. GO TO 1030
62 53 6l~
55
67 ..f>8 sg 70
·71 72 73 74 75 76 77 79 7S 30 8t 82 83 84 85 86 87 88 sg 9'i 91 92 g<
gr; 96 97 98 gg
100 101
.10 2 101 104
--105 106 107 108 iDS 110 111 112 11-::; 11L: 11 ::;: 11G
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FRITZ L480RATORY MATRIX PACKAGE CFLMXPK> ------------------------------- PAGE NO. 5 -------------------------------c c c
c c c c
1060
CHECK WHETHER THE DESIRED ACCURACY IS ~CHIEVEO.
IF <THRI.GE.THRF> GO TO 1020
RESET THRF TO BE IN READINESS FOR A SUBSEQUENT IMPROVEMENT PROCEDURE, IF ANY.
THRF=THRF~1.0~-6
RETURN END
117 118 119 120 121 122 123 12~
125 125 127
-------------------------------------------------------~-------------------SUBROUTINE GEVP CA,B,S,T,N>
c C ROUTINE TO SOLVE T~E GENERAL EIGENVALUE PROBLEM. C A~X=LAMBDA~B~X
c
c
COMMON /IYENGAR/ I,J,K,SUM,TEMP,YC9) REAL ACN,N),B(N,N),S(N,N),T(N)
C DIAGONALIZE MATRIX B. c
CALL EV (B,S,N> c C IMPROVEMENT PROCEDURE. c
CALL IEV CB,S,N> c C FORM THE MATRIX DROOUCT s~INVERSE OF G IN S. c
DO 1000 J=1 7 N TEMP=i.O/SQRTCB(J,J)) DO 1000 I=1,N
1000 S<I,J>=SCI,J>~TEMP c C MODIFY MATRIX A. c
CALL MULT CA,S,B,N,N,N> DO 1020 I=1,N DO 1020 J=I,N SUM=O.O DO 1010 K=1,N
1010 SUM=SUM+SCK,I>~BCK,J) A cr, J> =SUt1
1020 A(J,I>=SUM c C OBTAIN EIGENVALUES OF A. c
c
CALL EV (~,B,N)
CALL IEV CA,B,N)
C FORM THE MATRIX PRODUCT s~EIGENVECTO~ MATRIX C TO OBTAIN THE FINAL EIGENVECTORS. c
CALL POSTM CS,B,N,N,T) RETURN END
1 2 3 4 5 G 7 8 g
10 11 12 13 14 15 "' ,.. LO
17 18 19 20 21 22 23 2~
25 2S 27 28 29 30 31 32 33 34 35 36 37 38 39 4!J 41 42 lt 3 44 45
---------------------------------------------------------------------------
FRITZ L.t\BOR.ATORY MATRIX PACKAGE (FL~1XPK>
---------~------~-------------- PAGE NO. 6 -------------------------------· SUBROUTINE HINV (A,N,DET,NEXCHl 1
c 2 C ROUTINE TO INVERT MATRIX A. 3 c 4
CO~MON /IYENGAR/ BIG,I,J,K,NM!,NP!,NR,T,Y(6) 5 REAL A(N,N) 6 INTEGER NEXCH(N) 7
c 8 IF <N.NE.U GO TO 1000 9 DET=A(1,1) 10 A<1,1>=1.0/DET 11 RETURN 12
1000 DET=1.0 13 NM1=N-1 14 DO 1070 K=1,N 15
c 16 C SEARCH COLUMN 1 FOR THE LARGEST ABSOLUTE-VALUED 17 C ELEMENT IN ROWS K THROUGH N. 18 c 19
BIG=O.O 20 DO 1010 I=K,N 21 T=ABS<A<I,1)) 22 IF <BIG.GE.T> GO TO 1010 23 BIG=T 24
c 25 C THE LA~GEST ABSOLUTE-VALUED ELEMENT IS IN ROW NR. 26 c 27
NR=I 28 1010 CONTINUE 29
c 30 C EXCHANGE ROWS NR AND K, ONLY IF NR IS DIFFERENT FROM K. 31 c 32
IF CNR.EQ.K> GO TO 1030 33 DO 1020 J=1,N 34 T=A<K,Jl 3? ACK,J>=A<NR,J> 36
1020 A<NR,J>=T 37 c 38 C CHANGE THE SIGN OF THE DETERMINANT~ 39 C SINCE ROWS HAVE BEEN EXCHANGED. 40 c 41
DET=-DET 42 c 43 C BOOK-KEEPING VECTOR NEXCH STORES THE INFORMATION ~44 C ABOUT ROW EXCHANGES. 45 c 46
1030 NEXCHCK>=NR -47 T=ACK,1) '48
c C CONTINUOUS PART-PRODUCT FOR THE DETERMINANT.
49 50 51 52 53 54 55 56 57 58
c
c
DET=DET•T T=1.0/T
C MODIFY ROW K BY DIVISION BY THE PIVOT ELEMENT. c
DO 1040 J=1,NH1 1040 ACK,Jl=A<K,J+l>~T
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------------------------------- PAGE NO. 7 -------------------------------
c c G c
1050
1060 1070
c c c c c
c c c
c
1080 1090
1000
ACK,N>=T
MODIFY ALL ROWS OTHER THAN ROW K BY ADDITION OF SUITA9LE MULTIPLES OF ROW ~·
DO 1060 I=1,N IF CI.EQ.K> GO TO 1060 BIG=-A<I,1> DO 1050 J=1,NH1 ACI,J>=AC!,J+i)+BIG~ACK,JJ
A (I, N> =BIG~T CONTINUE CONTINUE
PER~ORM COLUMN EXC4ANGES TO MATCH PREVIOUS ROW ~XCHANGES
CIN REVERSE ORDER>. NOTE THAT NEXCHCN>=N, ALWAYS. HEN~€,
K IS NOT ALLOWED TO ASSUME THE VALUE N IN THE FOLLOWINS CO-LOOP.
NP1=N+1 00 1090 J=2,N K=NP1-J NR=NEXCHCK> IF CNR.EQ.K> GO TO 1090 DO 1080 I=1,N T=ACI,Kl ACI,K>=A<I,NR> A<I,NR>=T CONTINUE RETURN END
SUBROUTINE HOVE CA,B,H,N> _.;
ROUTINE TO COPY MATRIX A AS MATRIX B. O=A
,---coMr10N /IYENGAR/ I,MN,YC12) \
\REAL AC1),BC1J -. -HN=M~_N- '
I
DO 1000 __ I= 1 t.._M~~ j __., !'
B!I>=A<I> C' c /; RETURN END
5g so 51 52 63 64 65 6~ 57 6!3 E)g 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 135 87 88
-i _..
2 3 4 5 6 7 8 9
10 11 12
---------------------------------------------------------------------------
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FRITZ LABORATORY MATRIX PACKAGE <FV1XPK) ------------------------------- PAGE NO. 8 -------------------------------
c c c
c
1000 1010
SUSROUTIN~ HULT CA,G,C,L,H,N)
ROUTINE TO FORM THE PRODUCT OF MATRICES A AND B. C=A~B
COMMON /IYENGAR/ I,J,K,SUM,YC10) REAL ACL,H> ,BCH,tH ,CCL,Nl
DO 1010 I=1,L DO 1010 J=1,N SUN=O.O DO 1000 K=1,H SUM=SUM+ACI,K>~B<K,J)
CCI,J>=SUM RETURN END
1 2 3 4 5 6 7 8 g
·'UJ 11 12
"'13 14 15
---------------------------------------------------------------------------
c
SUBROUTINE OUTE <A,I,J,TITLE,TITEL> C011MON /IYENGAR/ II,IMAX,IROH,IV(2) ,I1,JCOL,JJ,.JMAX,J1,J2,KK,YC2l REAL A CI,J>
C ROUTINE TO PRINT MATRIX A IN E-FORMAT. C STORE E IN IVC2> FOR E-FORHAT OUTPUT. c
IV<2>=10HO{XE12.5}) c C JMAX = NUMBER OF VE~TICAL PARTITIONS OF THE MATRIX. c
1000 JMAX=CJ+9)/10 c C !MAX = NUMBER OF HORIZONTAL PARTITIONS OF THE MATRIX. c
c c c c
c c c c
c c c
c c
IHAX=<I+24)/25 IV<1>=10HC1HO,I4,X1 Y(1>=10H YC2>=10H 00 1020 JCOL=1 7 JMAX
PRESCRIBE THE FIRST (J1) AND LAST (JJ) COLUMN NUMBERS TO BE PRINTED ON THE PAGE.
JJ=10~JCOL
Ji=JJ-9 IF <J.LT.JJ) JJ=J 00 1020 IROW=i,IMAX
PRESCRIBE THE FIRST CI1l AND LAST <II> ROW NUMBERS TO BE PRINTED ON THE PAGE.
II=25li'IROW I1=II-24 IF CI.LT.II> II=I
LABEL TH~ OUTPUT, INDICATE MATRIX SIZE AND NUM9~R THE COLUMNS.
PRINT 1030, TITLE,TITEL,Y,I,J, CKK,KJ<:J1,JJ) 00 1010 KK-=!1,!!
NUMBE~ THE ROWS AND PRINT ELEMENTS OF THE HATRIX.
1 2 3 4 5 6 7 8 q
10 11 .. .... ..LC
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
- 2 3 29 30
-. 31 32 33 34 35 36 37 38 3g 40 41 42
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FRITZ LABO~ATORY MATRIX PACKAGE CFLMXPK) ------------------------------- PAGE NO. 9 -------------------------------C 43
1010 PRINT IV, KK,CACKK,J2),J2=J1,JJ) 44 YC1)=10H <CONTINUE 45
1020 Y (2) =10H0) 46 RETURN 47
c 48 c 49
ENTRY OUTF 50 c 51 C SUBROUTINE OUTF CA,I,J,TITLE,TITEL) 52 C ROUTINE TO PRINT MATRIX A IN F-FORMAT. 53 C STORE F IN IVC2) FO~ F-FORMAT OUTPUT. 54 c 55
IV<2>=10HO<XF12.5l) 56 GO TO 1000 57
c 58 c 59
ENTRY OUTG 61J c 61 C SUBROUTINE OUTG (A,I,J,TITLE,TITEL) 62 C ROUTINE TO PRINT MATRIX A IN G-FORMAT. 63 C STORE G IN IVC2> FOR G-FORMAT OUTPUT. 64 c 65
lVC2l=10HOCXG12.5l) 66 GO T0.1000 67
c 68 c 69
1030 FORMAT C1H110X4A10/1H010X4HSIZEI4,8H ROWS BYI4 7 8H COLUMNS/2HO 10!1 7C 13) 71
END 72
SUBROUTINE PMULT CA,B,K,L,X> c 2 C ROUTINE TO FORM THE MATRIX PRODUCT A~B IN A WHEN 8 IS SQUARE. 3 c 4
COMMON /IYENGAR/ I,J,M,SUM,Y<10l 5 REAL ACK,L>,BCL,U,XCL> 6
c 7 DO 1020 I=1 7 K ~
c 9 C GENERATE ELEMENTS OF I-TH ROW OF PRODUCT MATRIX 10 C AND STORE THESE IN VECTOR X. 11 c 12
00 1010 J=1 7 L 13 SUH=O.O 14 DO 1000 H=1,L 15
~000 SUM=SUM+ACI,Hl~B(M,J) 16 1010 XCJl=SUH 17
c 18 C REPLACE THE I-TH ~OW OF A BY THE VECTOR X. 19 c 20
DO 1020 J=1,L 21 1020 ACI,J>=XCJ> 22
RETURN 23 END 24
-------------------------~-------------------------------------------------
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FRITZ LABORATORY MATRIX PACKAGE CFLMXPK) ------------------------------- PAGE NO. 10 -------------------------------·
SUBROUTINE POSTM CA,B,K,L,X> 1 c 2 C ROUTINE TO FORM THE MATRIX PRODUCT A•B IN B WHEN A IS SQUARE. 3 c 4
COMMON /IYENGAR/ I,J,M,SUH,Y(10) 5 REAL ACK,K),B(K,U,XCK) 6
c 7 DO 1020 J=1,L 8
c .. g C GENERATE ELEMENTS OF J-TH COLUMN OF PRODUCT MAT~IX 10 C AND STORE THESE IN VECTOR X. 11 c 42
DO 1010 I=1,K 13 SUH=O.O 14 DO 1000 M=1,K 15
1000 SU~=SUM+ACI,M>•B<M,J) 16 1010 XCI>=SUM 17
c 18 C REPLACE THE J-TH COLUMN OF B BY VECTOR X. 19 c 20
c c c c
c
c c
c
00 1020 I=1,K 21 1020 BCI,J>=X<I> 22
RETURN 23 END 24
SUBROUTINE RDCBC CA,M,N)
~OUTINE TO READ I~ VALUES FOR ELEMENTS OF MATRIX A, COLUMN BY COLUMN.
COMMON /IYENGAR/ I,J,Y<12l REAL ACM,N)
DO 1000 J=1,N 1000 READ 1010, CACI,J>,I=1,H>
RETURN
1010 FORMAT C8F10.0) END
SUBROUTINE RDCOLG (A,H,N>
1 .2 3 4 5 6 7 8 g
10 11 12 13 14 15
C ROUTINE TO READ IN VALUES FOR ELEMENTS OF MATRIX A,
1 2
. 3 4 C COLUMN BY COLUMN IN A CONTINUOUS STRING OF COLUMNS.
c
c
c c
COMMON /IYENGAR/ YC14) REAL ACM,N)
REAO 1000, A RETURN
1000 FOR~AT C8F10.0) END
• 5 . 6
7 8 g
10 11 12 13 14
---------------------------------------------------------------------------
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FRITZ LA90~AT0RY MATRIX PACKAGE CFLMXPK) ------------------------------- PAGE NO. 11 -------------------------------
SUBROUTINE RDRBR CA,M,N) 1 c 2 C ROUTINE TO READ IN VALUES FOR ELEMENTS OF MATRIX A, 3 C ROW BY ROW. 4 c 5
c
c c
COMMON /IYENGAR/ I,J,Y{12> REAL ACM,N)
DO 1000 I=1,H 1000 READ 1010, CACI,Jl,J=1,N)
R.ETU RN
1010 FORMAT C8F10.0) END
5 7 3 g
10 11 12 13 !4 1?
---------------------------------------------------------------------------SUBqOUTINE RGROWG CA,M,N>
c C ROUTINE TO READ IN VALUES FOR ELEMENTS OF MATRIX C ROW BY ROW IN A CONTINUOUS STRING OF ROWS. c
c
c c
c
CO~MON /IYENGAR/ I,J,YC12> REAL ACM,N>
READ 1000, {{ACI,J>,J=1,N>,I=1,Hl RETURN
100Q FORMAT C8F10.0> END
SUBROUTINE SCMUL CA,M,N,X>
A,
C ROUTINE TO MULTIPLY MATRIX A BY THE SCALAR QUANTITY X. C A = X TIMES A c
c
COMMON /IYENGAR/ I,MN,YC12> REAL A(1}
HN=MJS.N DO 1000 I=1,MN
1000 ACI>=Xli-A(!) RETURN END
1 2 3 4 5 5 7 8 g
10 11 i2 13 14
1 2 3 4 5 5 7 8 g
10 11 12 13
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FRITZ LABORATORY MATRIX PACKAGE tFLMXPK) ------------------------------- PAGE NO. 12 -------------------------------
c c c c
c
c c c
c c c c c c c
1000
1010
1020
c c c
c c c
c c c c
c c c
1030
10·40
1050
SU9ROUTINE SINV <A,DA,N)
ROUTINE TO FINO THE INV~RSE OF A SYMMETRIC, POSITIVE-DEFINITE M~TRIX A.
COMMON /IYENGAR/ I,IFIN,INOEX,INIT,J,JP1,K,SUM,TEMP,YC5) REAL A< N, N)
IF (N.NE.U GO TO 1020 DA=A (1,1> A<1, 1>=1.0/DA RETURN
BUILT-IN SUBPROGRAM~
SUM=O.O DO 1010 K=INIT,IFIN SUM=SUMtACI,K)•ACJ,K) GO TO INDEX, <1040 7 1050,1070,1080)
GEN~RATE MATRIX LAMBDA CA=LAMBOA•LAMBOA TRANSPOS~>. NOTE - ONLY THE DIAGONAL ELEMENTS ARE THOSE OF LAMBDA-INVERSE-TRANSPOSE CL!T).
(1,1) ELEMENT OF LIT.
DA=1.0/SQRTCAt1,1)) AC1,1>=DA
FIRST COLUMN OF LAMBDA.
00 1030 I=2,N ACI,1>=AC1,I>•DA INIT=1 DO 1050 J=2,N ASSIGN 1040 TO INDEX IFIN=J-1 I=J GO TO 1000
J-TH DIAGONAL ELEMENT OF LIT <2 .u:. J .LE. N> •
TEMP=1.0/SQRTCACJ,J)-SUM) A CJ, J) =TEMP~
CONTINUOUS PART-PRODUCT FOR THE RECIPROCAL OF THE SQUARE ROOT OF THE DETERMINANT.
OA=DA""TEMP IF (J.EQ.N) GO TO 1050 ASSIGN 1050 TO INDEX JP1=J+1 00 1050 I=JP1,N GO TO 1000
J-TH COLUMN OF LAMBDA <2 .LE. J .LE. N-1).
ACI,Jl=tACJ,I)-SUH>""TEMP
1 2 3 4 5
7 8 g
" 10 11 12
..-11 14 15 16 17 18 19 20 21 22 23 24 25 25 27
29 30 31 32 33 34 35 35 37 38 39 40 41 42 43
- 44 ·4s
46 .. . 4 7
4.'3 4g 50 51 52 53 54 55 55 57 58
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FRITZ L~BORATORY MATRIX PAC~AGE CFLMXPK) ------------------------------- PAGE NO. 13 -------------------------------1060 ASSIGN 1070 TO INDEX 59
c so C FINAL VALUE OF THE DETERMINANT. 51 c 62
DA=D A~OA 6 3 OA=1. 0/DA 64
c 65 C GENERATE OFF-DIAGONAL ELEHENTS OF LIT. 66 c 67
DO 1070 I=2,N 68 IFIN=I-1 69 TEHP=A<I,I) 70 00 1070 J=1 1 IFIN 71 INIT=J 72 GO TO 1000 73
1070 ACJ,Il=-SUM~TEHP 74 ASSIGN 1080 TO INDEX 75 IFIN=N 76
c 77 C FOR~ THE PRODUCT LIT~INVERSE OF LAMBIJA. 78 C RESULT IS INVERSE OF THE ORIGINAL MATRIX A. 79 c 80
DO 1090 I=1,N 81 INIT=I 82 DO 1090 J=1,I 83 GO TO 1000 84
1080 ACI,J>=SUM 85 1090 A(J,I>=SUM ~6
RETURN 8 7 END 88
---------------------------------------------------------------------------c c c c
c
c
SUBROUTINE SOLVE CA,B,N,L,DET>
ROUTINE TO SOLVE THE SYSTEM OF LINEAR SIMULTANEOUS EQUATIONS A~X=B~
COMMON /IYENGAR/ A8SA,BIG,O,I,IJK,IP1,J,K,NM1,Y(4) ,z REAL ACN,N>,BCN,L>
IF {N.NE.U GO TO 1010 DET=A(1,1) DO 1000 J=1,L
1000 9C1,Jl=BC1,J)/DET RETURN
1010 Nt11=N-1 DET=1.0 DO 1070 I=1,NH1 IP1=I+1
C SEARCH COLUMN I FOR THE LARGEST ABSOLUTE-VALUED C ELEMENT IN ROWS I THROUGH N. c
c
BIG=O.O DO 1020 J=I,N ABSA=ABSCA<J,Il) IF <BIG.GE.ABSA) GO TO 1020 BIG=ABSA
1 2 3 4 5
7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
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~RITZ L~BORATORY HATRIX PACKAGE CFLMXPK) ------------------------------- PAGE NO. 14 -------------------------------c THE LARGEST ABSOLUTE-VALUED ELEMENT IS IN ROW K OF COLU~N I. 28 c
1020 c c c
1030
K=J CONTINUE
EXCHANGE ROWS K AND I, ONLY IF K IS DIFFERENT FROM I.
IF C K. EQ.!) GO TO 1050 DO 1030 J=I,N Z=ACI,J) A<I,J>=ACK,Jl ACK,J>=Z DO 1040 J=1,L Z=BCI,J> BCI,Jl=BCK,J)
1040 B<K,J>=Z c C CHANGE THE SIGN OF THE DETERMINANT, C SINCE ROWS HAVE BEEN EXCHANGED. c
DE T=-DET 1050 Z=ACI,I>
c C CONTINUOUS PART-PRODUCT FOR THE OETERMINANT. c
c c c c
c
DET=DET""Z Z=1.0/Z
MODIFY ELEMENTS OF A - THE REDUCTION PROCESS. MODIFY MATRIX 8 ALSO.
00 1070 K=IP1,N 0=-ACK,I>•Z 00 1060 J=i,L
1050 BCK,J>=B<K,J)+D•BCI,J) 00 1070 J=IPi,N
1070 ACK,J>=~(K,Jl+D•A(I,J) Z=ACN,N)
C FINAL VALUE OF THE ryETERMINANT. c
c
OET=DET• Z Z=1.0/Z
C PROCESS OF BACK-SUBSTITUTION TO GET THE SOLUTION MATRIX. c
00 1090 K=1,L BCN,K>=B<N,K)""Z 00 1090 IJK=1,NH1 I=N-IJK IP1=!+1 0=0.0 DO 1080 J=IP1,N
1080 O=O~ACI,J>•BCJ,Kl 1090 B<I,Kl=CBCI,Kl-0)/ACI,Il
RETURN END
29 30 31 32 33 34 35 35 .. 37 3~3"'
;39 40 41 4~ 43 44 45 46 47 48 49 51J 51 52 5? 54 55 55 57 58 5S 60 61 62 6 ",' 61, 55 66 67 68 69 70
' 71 72 73
-. 7 4 75 76 77 78 79 80 81 82 8:3 8Ll-
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FRITZ LABORATORY MATRIX PACKAGE (FL~XPKl
------------------------------- PAGE NO. 15 -------------------------------
c c c
c
c c c
SUBROUTINE SQTR (A,N) 1
ROUTINE TO TRANSPOSE THE SQUARE MATRIX A IN ITS OWN SPACE.
COMMON /IYENGAR/ I,INCR,K,L,LAST,NM1,NP1,T,Y(6) REAL A ( 1)
IF <N.EQ.1) RETURN NN1=N-1 NP1=N+1 LAST:::N.v-N INCR=O DO 1000 K=2,N LAST=LAST-N INC~=INCR+NM1 00 10GO L=K,LAST,NP1 I=L+INCR.
EXCHANGE ELEMENTS WITH SUBSCRIPTS I AND L.
2 3 i+ 5 6 7 8 g
10 _,, . ... .:..
12 13 14 15 16 17 18 1q
1000
T= A (I)
ACH=A<L> A<U =T Rt:TURN END
20 21 22 23 2i+ 25
c c c
c
SUBROUTINE SUB CA,9,C,M,N)
ROUTINE TO SUBTRACT MATRIX B FROM MATRIX Ao C=A-B
COMMON /IYENGAR/ I,MN,Y(12) REAL A<1>,B<U,C(1)
MN=M•N DO 1000 I=1,MN
1000 CCI>=A<I>-BCI) RETURN END
1
:-~
4 5 6 7
8 g
10 11 12
---------------------------------------------------------------------------. ·.·~ .. S'UBROUTINE TMULT CA,B,C,L,H,N) 1
c 2 C ROUTINE TO FORM THE MATR.IX PRODUCT OF <TRANSPOSE OF MATRIX A> 3 C AND MATRIX B. C=<TRANSPOSE OF A>•B 4 c 5
COMMON /IYENGAR/ I,J,K,SUM,Y(10) 6 REAL ACL,Ml,BCL,Nl,CCM,Nl 7
c 8 DO 1010 I=1,M 9 DO 1010 J=1,N 10 SUM=O.O 11 DO 1000 K=1,L 12
1000 SUM=SUM+A(K,Il~BcK,J) 13 1010 C<I,Jl=SUM 14
RETURN 15 END 16
---------------------------------------------------------------------------
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FRITZ LABORATORY MATRIX PACKAGE CFL~XPK)
------------------------------- PAGE NO. 15 -------------------------------SUB~OUTINE TRANS CA,B,~,N> 1
c 2 C ROUTINE TO TRANSPOSE MATRIX A. B=TRANSPOSE OF A 3 c 4
COMMON /IYENGAR/ I,J,YC12l 5 REAL A(H,N),B(N,M) 5
c 7 00 1000 I=1,M DO 1000 J=!,N
1000 BCJ,I>=ACI,J> RETURN END
g .. 10
11 .--12
---------------------------------------------------------------------------SUBP.OUTINE XABATC CA,B,C,L,M,X) c C ROUTINE TO FORM THE ~ATRIX PRODUCT C A~B•<TRANSPOSE OF A> WHEN 8 IS SYMMETRIC. c
c
c
COMMON /IYENGAR/ r,J,K,SUM,YC10) REAL ACL,M> ,BCM,M) ,CCL,L) ,XCM)
DO 1030 I=1,L
C GENERATE ROW I OF PART-PRODUCT A~B IN VECTOR X. c
c
DO 1010 J=1,M SUM=O.O DO 1000 K=!,M
1000 SUM=SUH+A(I,K>~B(K,J> 1010 XCJ>=SUH
C PRODUCT MATRIX C IS SYMMETRIC, SINCE MATRIX 8 IS SYMMETRIC. C GENeRATE PRODUCT HAT~IX. c
00 1030 J=I,L SUM=O.O 00 1020 K=1,M
1020 SUM=SUM+X(K}~ACJ,K) C CJ, Il =SUM
1030 C<I,J>=SUM RETURN END
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10 11 12 13 14 15 15 17 13 19 20 21 22 23 24 25 26 27 28 29
---------------------------------------------------------------------------
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FRITZ LABORATORY MATRIX PACKAGE CFL~XPK> ------------------------------- PAGE NO. 17 ------------------------------. SUBROUTINE XABTA CA,B,L,N,X) c C ROUTINE TO FORM TY~ MATRIX PRODUCT C A'~-CTRANSPOSE OF Bl IN A WHEN B IS SQUARE. c
COMMON /IYENGAR/ I,J,K,SUM,YC10) REAL ACL,Nl,BCN,Nl,XCN>
c
c C GENERATE ROW I OF PRODUCT MATRIX IN VECTOR X. c
DO 1010 J=1,N SUM=O.O DO 1000 K=1,N
1000 SUM=SUM~ACI,K>•B<J,K) 1010 XCJ>=SUM
c C REPLACE ROW I OF MATRIX A BY VECTOR x. c
DO 1020 J=1,N 1020 ACI,Jl=X(J)
RETURN END
E I (
<::
1C 11 12 13 14 .15 16 17 113 1S 20 21 22 23 24----------------------------------------------------------------------------SUSqOUTINE XABTC CA~B,C,L,M,N>
c C ROUTINE TO FORM THE MATRIX PRODUCT C A4 CTRANSPOSE OF 8) c
c COMMON /IYENGAR/ r,J,K,SUM,YC10) REAL ACL,M),3CN,M),CCL,N)
DO 1010 I=1,L DO 1010 J=1,N SUM=O.O 00 1000 K=1,M
1000 SUM=SUM+ACI,K>•BCJ,K) 1 0 1 0 c (I, J) = su ti
RETURN END
1 2 3 4 5 5 7 8 g
10 11 12 13 14 1'5 15 ---------------------------------------------------------------------------
--···-u.--~-
. 4 FRITZ L4BO~ATORY MATRIX PAC~~GE CFLMXPK> • ------------------------------- PAGE NO. 18 -------------------------------
)
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SUBROUTINE XATBAC CA,B,C,L,M,X> 1 2·
ROUTINE TO FORM THE MATRIX PRODUCT 3 <TRANSPOSE OF A>•A•A WHEN B IS SYMMETRIC. 4
c c c
5 c COMMON /IYENGAR/ I,J,K,SUH,Y(10) G REAL A<L,Ml,BCL,Ll,CCM,H>,XCL) 7
8 c DO 1030 J=1,M 9 ,0
11 12 f3 14 15 15 17' 18 19 20 21 22 23 24 25 26 27 2e 29
c c c
c c c c
1000 1010
GENERATE COLUMN J OF PART-PRODUCT s•A IN VECTOR X.
DO 1010 I=1,L SUH=O.O DO 1000 K=1,L SUH=SUM+BCI,K>•ACK,Jl X<I>=SUH
PRODUCT MATRIX C IS SY~METRIC, SINCE HATR!X B IS SYMM~TRIC. GENERATE PRODUCT MATRIX.
DO 1030 I=1,J SUM= 0. 0 DO 1020 K=1,L
1020 SUM=SUM+ACK,I>•X<K> C (!, Jl =SUt1
1030 C(J,Il=SUM RETURN END
---------------------------------------------------------------------------c C ROUTINE TO FORM TH~ MATqix PRODUCT C <TRANSPOSE OF A>•B IN B WHEN A IS S~UARE. c
c
c
COMMON /IYENGAR/ I,J,K,SUM,YC10) REAL ACL,L> ,B<L,N> ,XCL)
00 1020 J=1,N
C GENERATE COLUMN J OF PRODUCT MATRIX IN VECTOR X. c "
c
00 1010 I=i,L SUM=O.O DO 1000 K=1,L
1000 SUM=SUM+ACK,I>•BCK,J) 1010 XC!)=SUM
C REPLACE COLUMN J OF MATRIX B BY VECTOR X. c
DO 1020 I=1,L 1020 B(!,J):X(Il
RETURN END
1 2 3 4 5· 6~ 7 8 9
10 11 12 13
'14 ~15
16 <17
18 19 20 21 22 23 24
---------------------------------------------------------------------------
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ACKNO\'lLEDG.MENTS
This documentation has been prepared at the Fritz
Engineering Laboratory, Department of Civil Engineering,
Lehigh University, Bethlehem, Pennsylvania. Dr. David A.
VanHorn is the Chairman of the Department, and Dr. Lynn s.
Beedle is the Director of the Laboratory.
By his appointment as a member of the Computer
Systems Group in Fall 1968, Hr. Iyengar had the opportunity
to peruse and debug several programs written by fellow
research workers in the Laboratory. Dr. Celal N. Kostem
is the Chairman of the Group.
Many colleagues have contributed useful suggestions
which now form parts of this work. Mr. Ed\vard T ~ Manning,
Jr. was associated with Mr. Iyengar in the development of
the earlier version (FCMXPK} of this package.
A special debt of gratitude is due to Mrs. Michele
Kostem who patiently read through the manuscript and typed
the draft copy.
Sincere appreciation is extended to Mrs. Mary C.
Ambrose who typed the final copy, assisted by Mrs. Ruth
A. Grimes.
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REFERENCES
1. Ralston, Anthony and \'lilf, Herbert s. (Eds) MATHEHATICAL NETHODS FOR DIGITAL COMPUTERS John Wiley and Sons, Inc., New York, 1960
2. Control Data Corporation CONTROL DATA 6000 SERIES CO~WUTER SYSTEMS SCOPE 3.3 REFERENCE MANUAL (60305200B) Sunnyvale, California, 1971
3. Lehigh University Computing Center COMPUTING NOTICE, APRIL-MAY 1971 Bethlehem, Pennsylvania, Nay 1971
4. Wylie, C. R., Jr. ADVANCED ENGINEERING MATH~~TICS IvlcGraH-Hill Book Company New York, 1965
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